LMFDB - L-function 2-40e2-40.29-c1-0-31

Dirichlet series

L(s) = 1

+ 2·3^-s + 9^-s − 6i·11^-s − 6i·17^-s − 2i·19^-s − 4·27^-s − 12i·33^-s − 6·41^-s + 10·43^-s + 7·49^-s − 12i·51^-s − 4i·57^-s + 6i·59^-s + 14·67^-s − 2i·73^-s + ⋯

L(s) = 1

+ 1.15·3^-s + 0.333·9^-s − 1.80i·11^-s − 1.45i·17^-s − 0.458i·19^-s − 0.769·27^-s − 2.08i·33^-s − 0.937·41^-s + 1.52·43^-s + 49^-s − 1.68i·51^-s − 0.529i·57^-s + 0.781i·59^-s + 1.71·67^-s − 0.234i·73^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1600$ = $2^{6} \cdot 5^{2}$
Sign:	$0.316 + 0.948i$
Analytic conductor:	$12.7760$
Root analytic conductor:	$3.57436$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1600} (1249, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1600,\ (\ :1/2),\ 0.316 + 0.948i)$

Particular Values

$L(1)$	$\approx$	$2.252996842$
$L(\frac12)$	$\approx$	$2.252996842$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	3	$ 1 - 2T + 3T^{2} $
	7	$ 1 - 7T^{2} $
	11	$ 1 + 6iT - 11T^{2} $
	13	$ 1 + 13T^{2} $
	17	$ 1 + 6iT - 17T^{2} $
	19	$ 1 + 2iT - 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 37T^{2} $
	41	$ 1 + 6T + 41T^{2} $
	43	$ 1 - 10T + 43T^{2} $
	47	$ 1 - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 - 6iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 14T + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 2iT - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 - 18T + 83T^{2} $
	89	$ 1 + 18T + 89T^{2} $
	97	$ 1 - 10iT - 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−9.043286683129683219459596029754, −8.591579832571673505528003084598, −7.82336287633584810161167813566, −7.02556548992560164162144608904, −5.96858853173716671788493034377, −5.14925602938534690765238908263, −3.89753377453048086960262528029, −3.08735359786260095590437152702, −2.43016051313926583327314117215, −0.74244896642002666158509946344, 1.71493873068809262695085591796, 2.42481699904604462119437761230, 3.64768968698289207823728794986, 4.29569294913590630432367766231, 5.41852473123892389915316951896, 6.49255958603362394750607504820, 7.38715341089691132133790223606, 8.019838620753455863560859146050, 8.730054488693157569360021493151, 9.534829036418252700257629357443

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.252996842\)
\(L(\frac12)\)	\(\approx\)	\(2.252996842\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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