LMFDB - L-function 2-3276-13.12-c1-0-35

Dirichlet series

L(s) = 1

− 0.257i·5^-s − i·7^-s − 4.19i·11^-s + (0.742 − 3.52i)13^-s − 7.46·17^-s − 6.33i·19^-s − 3.93·23^-s + 4.93·25^-s − 3.63·29^-s + 8.55i·31^-s − 0.257·35^-s + 5.43i·37^-s + 8.29i·41^-s − 6.23·43^-s + 9.91i·47^-s + ⋯

L(s) = 1

− 0.115i·5^-s − 0.377i·7^-s − 1.26i·11^-s + (0.205 − 0.978i)13^-s − 1.80·17^-s − 1.45i·19^-s − 0.820·23^-s + 0.986·25^-s − 0.674·29^-s + 1.53i·31^-s − 0.0435·35^-s + 0.892i·37^-s + 1.29i·41^-s − 0.950·43^-s + 1.44i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3276$ = $2^{2} \cdot 3^{2} \cdot 7 \cdot 13$
Sign:	$-0.978 - 0.205i$
Analytic conductor:	$26.1589$
Root analytic conductor:	$5.11458$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3276} (2521, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3276,\ (\ :1/2),\ -0.978 - 0.205i)$

Particular Values

$L(1)$	$\approx$	$0.4266792702$
$L(\frac12)$	$\approx$	$0.4266792702$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	7	$ 1 + iT $
	13	$ 1 + (-0.742 + 3.52i)T $
good	5	$ 1 + 0.257iT - 5T^{2} $
	11	$ 1 + 4.19iT - 11T^{2} $
	17	$ 1 + 7.46T + 17T^{2} $
	19	$ 1 + 6.33iT - 19T^{2} $
	23	$ 1 + 3.93T + 23T^{2} $
	29	$ 1 + 3.63T + 29T^{2} $
	31	$ 1 - 8.55iT - 31T^{2} $
	37	$ 1 - 5.43iT - 37T^{2} $
	41	$ 1 - 8.29iT - 41T^{2} $
	43	$ 1 + 6.23T + 43T^{2} $
	47	$ 1 - 9.91iT - 47T^{2} $
	53	$ 1 + 1.63T + 53T^{2} $
	59	$ 1 - 2.72iT - 59T^{2} $
	61	$ 1 + 4.54T + 61T^{2} $
	67	$ 1 + 5.84iT - 67T^{2} $
	71	$ 1 + 2.05iT - 71T^{2} $
	73	$ 1 - 13.9iT - 73T^{2} $
	79	$ 1 + 0.0979T + 79T^{2} $
	83	$ 1 + 10.3iT - 83T^{2} $
	89	$ 1 + 4.60iT - 89T^{2} $
	97	$ 1 + 10.7iT - 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

−8.513965979552877327849077797193, −7.51008144219460891398072993009, −6.63740739323350107229419175433, −6.14330176090052074560079530896, −5.04926989913477320103263758639, −4.49767788954929080250343359117, −3.32089168728891205471997290263, −2.72928974954659336799779295943, −1.28610722456293395051990019899, −0.12690236858152335507266055452, 1.91519043399298284939048364015, 2.20056182311185154659689413356, 3.78467211047888398190177503111, 4.28296241601645308377711384704, 5.21075288437793978838093397227, 6.14977817920149072800704397075, 6.78986228817700037702550278460, 7.47594763328134768990965321619, 8.357440473233158882143378668977, 9.092505845228372312571938449566

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4266792702\)
\(L(\frac12)\)	\(\approx\)	\(0.4266792702\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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