LMFDB - L-function 2-3380-13.12-c1-0-27

Dirichlet series

L(s) = 1

− 3·3^-s + i·5^-s − 3i·7^-s + 6·9^-s − 3i·11^-s − 3i·15^-s + 7·17^-s + i·19^-s + 9i·21^-s + 7·23^-s − 25^-s − 9·27^-s − 5·29^-s − 4i·31^-s + 9i·33^-s + ⋯

L(s) = 1

− 1.73·3^-s + 0.447i·5^-s − 1.13i·7^-s + 2·9^-s − 0.904i·11^-s − 0.774i·15^-s + 1.69·17^-s + 0.229i·19^-s + 1.96i·21^-s + 1.45·23^-s − 0.200·25^-s − 1.73·27^-s − 0.928·29^-s − 0.718i·31^-s + 1.56i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3380$ = $2^{2} \cdot 5 \cdot 13^{2}$
Sign:	$0.554 + 0.832i$
Analytic conductor:	$26.9894$
Root analytic conductor:	$5.19513$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3380} (3041, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3380,\ (\ :1/2),\ 0.554 + 0.832i)$

Particular Values

$L(1)$	$\approx$	$1.006900321$
$L(\frac12)$	$\approx$	$1.006900321$
$L(\frac{3}{2})$		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 - iT $
	13	$ 1 $
good	3	$ 1 + 3T + 3T^{2} $
	7	$ 1 + 3iT - 7T^{2} $
	11	$ 1 + 3iT - 11T^{2} $
	17	$ 1 - 7T + 17T^{2} $
	19	$ 1 - iT - 19T^{2} $
	23	$ 1 - 7T + 23T^{2} $
	29	$ 1 + 5T + 29T^{2} $
	31	$ 1 + 4iT - 31T^{2} $
	37	$ 1 - 3iT - 37T^{2} $
	41	$ 1 - 7iT - 41T^{2} $
	43	$ 1 - 9T + 43T^{2} $
	47	$ 1 + 8iT - 47T^{2} $
	53	$ 1 + 6T + 53T^{2} $
	59	$ 1 + 5iT - 59T^{2} $
	61	$ 1 + 5T + 61T^{2} $
	67	$ 1 - 13iT - 67T^{2} $
	71	$ 1 + 3iT - 71T^{2} $
	73	$ 1 - 14iT - 73T^{2} $
	79	$ 1 + 8T + 79T^{2} $
	83	$ 1 - 12iT - 83T^{2} $
	89	$ 1 + 7iT - 89T^{2} $
	97	$ 1 + 11iT - 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

−8.313575066734613070156840536951, −7.35594269230784983402777603250, −7.07369675414016349169216927033, −6.04014989213173529271941018494, −5.67378014174418122355350416254, −4.80746468223207447065954966798, −3.91866914276414711119124742205, −3.11246541908254669871321818838, −1.30561566825963211679558134676, −0.59498162436130934325145437729, 0.864693575847817334454316249255, 1.88169768979696503699066199838, 3.23339388130858337175357263748, 4.50739656012929874061090337443, 5.09963019013441882214160535541, 5.65484072617136688869015866072, 6.19640453111125923135565079668, 7.23444124905394986400759370423, 7.68478546185792904855775722537, 9.043851085202135989293447394655

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.006900321\)
\(L(\frac12)\)	\(\approx\)	\(1.006900321\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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