LMFDB - L-function 2-40e2-40.29-c1-0-17

Dirichlet series

L(s) = 1

− 3.44·3^-s + 8.89·9^-s − 5.44i·11^-s + 1.89i·17^-s + 6.34i·19^-s − 20.3·27^-s + 18.7i·33^-s − 6.79·41^-s + 10·43^-s + 7·49^-s − 6.55i·51^-s − 21.8i·57^-s − 6i·59^-s + 0.348·67^-s − 15.6i·73^-s + ⋯

L(s) = 1

− 1.99·3^-s + 2.96·9^-s − 1.64i·11^-s + 0.460i·17^-s + 1.45i·19^-s − 3.91·27^-s + 3.27i·33^-s − 1.06·41^-s + 1.52·43^-s + 49^-s − 0.917i·51^-s − 2.90i·57^-s − 0.781i·59^-s + 0.0425·67^-s − 1.83i·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

$L(1)$	$\approx$	$0.6653378516$
$L(\frac12)$	$\approx$	$0.6653378516$
$L(\frac{3}{2})$		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 + 3.44T + 3T^{2} $
	7	$ 1 - 7T^{2} $
	11	$ 1 + 5.44iT - 11T^{2} $
	13	$ 1 + 13T^{2} $
	17	$ 1 - 1.89iT - 17T^{2} $
	19	$ 1 - 6.34iT - 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 37T^{2} $
	41	$ 1 + 6.79T + 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 - 0.348T + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 15.6iT - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 6.55T + 83T^{2} $
	89	$ 1 - 4.10T + 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

−9.478104198657370703627850564421, −8.356791707542443650915118236906, −7.48790602098955109167051639975, −6.50197198095000856121783184517, −5.88102577926111353205268279583, −5.47403497463139711757498571672, −4.36192491862872156380671793395, −3.50767862226550132018734544717, −1.59414790499809998539044405142, −0.45991367095403731573251505221, 0.936932629245459087448355313027, 2.25259962310594989455302925968, 4.11626779992499838603258934124, 4.77425300252919880412751809457, 5.36739073328481432068376946444, 6.34186270191794645930235771020, 7.11413073637574541976050519380, 7.42324724046062831726326837877, 9.042127458443943868592243288390, 9.847540903722436454100060202775

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6653378516\)
\(L(\frac12)\)	\(\approx\)	\(0.6653378516\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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