LMFDB - L-function 16-2736e8-1.1-c1e8-0-2

Dirichlet series

L(s) = 1

− 4·19^-s + 16·25^-s − 32·29^-s + 24·41^-s + 28·43^-s − 26·49^-s − 8·53^-s + 8·59^-s − 8·61^-s − 24·71^-s + 4·73^-s − 16·89^-s − 8·107^-s − 24·113^-s + 56·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 16·169^-s + 173^-s + 179^-s + ⋯

L(s) = 1

− 0.917·19^-s + 16/5·25^-s − 5.94·29^-s + 3.74·41^-s + 4.26·43^-s − 3.71·49^-s − 1.09·53^-s + 1.04·59^-s − 1.02·61^-s − 2.84·71^-s + 0.468·73^-s − 1.69·89^-s − 0.773·107^-s − 2.25·113^-s + 5.09·121^-s + 0.0887·127^-s + 0.0873·131^-s + 0.0854·137^-s + 0.0848·139^-s + 0.0819·149^-s + 0.0813·151^-s + 0.0798·157^-s + 0.0783·163^-s + 0.0773·167^-s + 1.23·169^-s + 0.0760·173^-s + 0.0747·179^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{32} \cdot 3^{16} \cdot 19^{8}$
Sign:	$1$
Analytic conductor:	$5.18973\times 10^{10}$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{32} \cdot 3^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$

Particular Values

$L(1)$	$\approx$	$0.5891916076$
$L(\frac12)$	$\approx$	$0.5891916076$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	19	$ 1 + 4 T + 4 T^{2} - 4 p T^{3} - 362 T^{4} - 4 p^{2} T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} $
good	5	$ 1 - 16 T^{2} + 161 T^{4} - 1184 T^{6} + 6544 T^{8} - 1184 p^{2} T^{10} + 161 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} $
	7	$ ( 1 + 13 T^{2} + 16 T^{3} + 80 T^{4} + 16 p T^{5} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} $
	11	$ 1 - 56 T^{2} + 1433 T^{4} - 23296 T^{6} + 285520 T^{8} - 23296 p^{2} T^{10} + 1433 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} $
	13	$ 1 - 16 T^{2} + 236 T^{4} - 4656 T^{6} + 70854 T^{8} - 4656 p^{2} T^{10} + 236 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} $
	17	$ 1 - 112 T^{2} + 5825 T^{4} - 182944 T^{6} + 3786816 T^{8} - 182944 p^{2} T^{10} + 5825 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} $
	23	$ 1 - 120 T^{2} + 6996 T^{4} - 262632 T^{6} + 7039718 T^{8} - 262632 p^{2} T^{10} + 6996 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} $
	29	$ ( 1 + 16 T + 176 T^{2} + 1328 T^{3} + 8206 T^{4} + 1328 p T^{5} + 176 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} $
	31	$ 1 - 32 T^{2} + 2156 T^{4} - 90400 T^{6} + 2569126 T^{8} - 90400 p^{2} T^{10} + 2156 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} $
	37	$ 1 - 112 T^{2} + 5740 T^{4} - 210384 T^{6} + 7515782 T^{8} - 210384 p^{2} T^{10} + 5740 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} $
	41	$ ( 1 - 12 T + 88 T^{2} - 228 T^{3} + 654 T^{4} - 228 p T^{5} + 88 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} $
	43	$ ( 1 - 14 T + 221 T^{2} - 1822 T^{3} + 15292 T^{4} - 1822 p T^{5} + 221 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} $
	47	$ 1 - 264 T^{2} + 33881 T^{4} - 2754624 T^{6} + 154366144 T^{8} - 2754624 p^{2} T^{10} + 33881 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} $
	53	$ ( 1 + 4 T + 120 T^{2} + 572 T^{3} + 8382 T^{4} + 572 p T^{5} + 120 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} $
	59	$ ( 1 - 4 T + 108 T^{2} - 516 T^{3} + 9110 T^{4} - 516 p T^{5} + 108 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} $
	61	$ ( 1 + 4 T + 105 T^{2} + 464 T^{3} + 10344 T^{4} + 464 p T^{5} + 105 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} $
	67	$ 1 - 328 T^{2} + 50588 T^{4} - 5003576 T^{6} + 373708006 T^{8} - 5003576 p^{2} T^{10} + 50588 p^{4} T^{12} - 328 p^{6} T^{14} + p^{8} T^{16} $
	71	$ ( 1 + 12 T + 252 T^{2} + 2236 T^{3} + 25958 T^{4} + 2236 p T^{5} + 252 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} $
	73	$ ( 1 - 2 T + p T^{2} - 154 T^{3} + 1156 T^{4} - 154 p T^{5} + p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} $
	79	$ 1 - 352 T^{2} + 69484 T^{4} - 8966112 T^{6} + 834290726 T^{8} - 8966112 p^{2} T^{10} + 69484 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} $
	83	$ 1 - 328 T^{2} + 47028 T^{4} - 4160408 T^{6} + 323364230 T^{8} - 4160408 p^{2} T^{10} + 47028 p^{4} T^{12} - 328 p^{6} T^{14} + p^{8} T^{16} $
	89	$ ( 1 + 8 T + 128 T^{2} + 984 T^{3} + 12830 T^{4} + 984 p T^{5} + 128 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} $
	97	$ 1 - 504 T^{2} + 115196 T^{4} - 16586824 T^{6} + 1793017734 T^{8} - 16586824 p^{2} T^{10} + 115196 p^{4} T^{12} - 504 p^{6} T^{14} + p^{8} T^{16} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.65479114581979646971640782261, −3.49285844945708422297380120312, −3.46376255144391872389964437439, −3.36074612651371626150426550222, −3.25411463975647464239616573939, −2.94072726966695542588225919662, −2.89372062001861472549384992964, −2.81526158777226776258016052358, −2.77163999627394706853714554938, −2.59368658166077178665789697429, −2.55063481445263247448761028412, −2.31749561623947648162121564193, −2.09220117641485757543390499536, −2.01286316225656687623527404960, −1.90065045273467797391352976594, −1.88609722570786877711284702513, −1.57110586327258202307608169211, −1.41741557647705570143685353113, −1.38234671461081586838014478821, −1.19866591368158972790674364463, −1.04020839903903565628283428557, −0.56784824311960904935874382823, −0.56368098727671031124915985468, −0.54239021012702089860349712396, −0.06194571828136504066417995844, 0.06194571828136504066417995844, 0.54239021012702089860349712396, 0.56368098727671031124915985468, 0.56784824311960904935874382823, 1.04020839903903565628283428557, 1.19866591368158972790674364463, 1.38234671461081586838014478821, 1.41741557647705570143685353113, 1.57110586327258202307608169211, 1.88609722570786877711284702513, 1.90065045273467797391352976594, 2.01286316225656687623527404960, 2.09220117641485757543390499536, 2.31749561623947648162121564193, 2.55063481445263247448761028412, 2.59368658166077178665789697429, 2.77163999627394706853714554938, 2.81526158777226776258016052358, 2.89372062001861472549384992964, 2.94072726966695542588225919662, 3.25411463975647464239616573939, 3.36074612651371626150426550222, 3.46376255144391872389964437439, 3.49285844945708422297380120312, 3.65479114581979646971640782261

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5891916076\)
\(L(\frac12)\)	\(\approx\)	\(0.5891916076\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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