LMFDB - L-function 2-1350-5.4-c3-0-71

Dirichlet series

L(s) = 1

− 2i·2^-s − 4·4^-s − 29.5i·7^-s + 8i·8^-s − 46.5·11^-s − 92.0i·13^-s − 59.0·14^-s + 16·16^-s + 4.53i·17^-s − 87.5·19^-s + 93.0i·22^-s + 160. i·23^-s − 184.·26^-s + 118. i·28^-s − 241.·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 1.59i·7^-s + 0.353i·8^-s − 1.27·11^-s − 1.96i·13^-s − 1.12·14^-s + 0.250·16^-s + 0.0647i·17^-s − 1.05·19^-s + 0.901i·22^-s + 1.45i·23^-s − 1.38·26^-s + 0.797i·28^-s − 1.54·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1350$ = $2 \cdot 3^{3} \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$79.6525$
Root analytic conductor:	$8.92482$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1350} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1350,\ (\ :3/2),\ 0.447 - 0.894i)$

Particular Values

$L(2)$	$\approx$	$0.1953243093$
$L(\frac12)$	$\approx$	$0.1953243093$
$L(\frac{5}{2})$		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 + 2iT $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 + 29.5iT - 343T^{2} $
	11	$ 1 + 46.5T + 1.33e3T^{2} $
	13	$ 1 + 92.0iT - 2.19e3T^{2} $
	17	$ 1 - 4.53iT - 4.91e3T^{2} $
	19	$ 1 + 87.5T + 6.85e3T^{2} $
	23	$ 1 - 160. iT - 1.21e4T^{2} $
	29	$ 1 + 241.T + 2.43e4T^{2} $
	31	$ 1 + 2.68T + 2.97e4T^{2} $
	37	$ 1 + 20.6iT - 5.06e4T^{2} $
	41	$ 1 - 501.T + 6.89e4T^{2} $
	43	$ 1 + 294. iT - 7.95e4T^{2} $
	47	$ 1 + 478. iT - 1.03e5T^{2} $
	53	$ 1 + 243. iT - 1.48e5T^{2} $
	59	$ 1 + 383.T + 2.05e5T^{2} $
	61	$ 1 - 132.T + 2.26e5T^{2} $
	67	$ 1 - 582. iT - 3.00e5T^{2} $
	71	$ 1 - 566.T + 3.57e5T^{2} $
	73	$ 1 - 839. iT - 3.89e5T^{2} $
	79	$ 1 + 451.T + 4.93e5T^{2} $
	83	$ 1 + 301. iT - 5.71e5T^{2} $
	89	$ 1 - 739.T + 7.04e5T^{2} $
	97	$ 1 + 1.14e3iT - 9.12e5T^{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

−8.512251218651421751541729911104, −7.67965959235571825656848035582, −7.32393548612553855961698234348, −5.79321996217735920997076546791, −5.15239751428240209354505118023, −3.98359432289464594331800742606, −3.35244264243805803148364687405, −2.20507243509820127050795488789, −0.837466679985845085924635126832, −0.05523780219833771767436212974, 1.97153335911793548683504546136, 2.70064559782080553064572950336, 4.26649619733231476727347546058, 4.92382291652124172997098713968, 5.99085261939686463312898525102, 6.38994778471479198009879329820, 7.53811911762226991541906710173, 8.287051936291418077199311548307, 9.147079827447861197473349222281, 9.393114681324644429363752571862

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.1953243093\)
\(L(\frac12)\)	\(\approx\)	\(0.1953243093\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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