LMFDB - L-function 2-40e2-8.5-c1-0-17

Dirichlet series

L(s) = 1

+ 1.44i·3^-s + 0.898·9^-s + 0.550i·11^-s + 7.89·17^-s − 8.34i·19^-s + 5.65i·27^-s − 0.797·33^-s + 12.7·41^-s + 10i·43^-s − 7·49^-s + 11.4i·51^-s + 12.1·57^-s − 6i·59^-s + 14.3i·67^-s + 13.6·73^-s + ⋯

L(s) = 1

+ 0.836i·3^-s + 0.299·9^-s + 0.165i·11^-s + 1.91·17^-s − 1.91i·19^-s + 1.08i·27^-s − 0.138·33^-s + 1.99·41^-s + 1.52i·43^-s − 49^-s + 1.60i·51^-s + 1.60·57^-s − 0.781i·59^-s + 1.75i·67^-s + 1.60·73^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1600$ = $2^{6} \cdot 5^{2}$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$12.7760$
Root analytic conductor:	$3.57436$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1600} (801, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1600,\ (\ :1/2),\ 0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.907299712$
$L(\frac12)$	$\approx$	$1.907299712$
$L(\frac{3}{2})$		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 - 1.44iT - 3T^{2} $
	7	$ 1 + 7T^{2} $
	11	$ 1 - 0.550iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 7.89T + 17T^{2} $
	19	$ 1 + 8.34iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 - 12.7T + 41T^{2} $
	43	$ 1 - 10iT - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 6iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 14.3iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 13.6T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 11.4iT - 83T^{2} $
	89	$ 1 + 13.8T + 89T^{2} $
	97	$ 1 - 10T + 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

−9.619857404528241626741842158668, −8.946481254334705570914608073728, −7.85831470571057881104171541310, −7.21938286544327977476481950636, −6.19968026574340729958235694751, −5.18327542363076171924691428659, −4.57464141933300830044398006148, −3.59542219687515530553426964008, −2.67765571868344189571972371844, −1.07943086614518558522052759437, 0.998474776735294409575130738536, 1.94167666887771177989016513276, 3.27683334603512466694953142117, 4.13279344236918938774005565884, 5.46933108977515997538422189439, 6.02812811742312240303260625299, 7.00451712624777981599425233028, 7.84792396343768148948652194813, 8.110494225206469032251089054271, 9.416299953616083568287370933279

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.907299712\)
\(L(\frac12)\)	\(\approx\)	\(1.907299712\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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