LMFDB - L-function 2-3100-31.30-c2-0-8

Dirichlet series

L(s) = 1

− 3.39i·3^-s − 1.77·7^-s − 2.55·9^-s − 7.22i·11^-s + 15.0i·13^-s + 10.2i·17^-s − 4.10·19^-s + 6.04i·21^-s − 22.1i·23^-s − 21.9i·27^-s + 23.4i·29^-s + (15.2 − 27.0i)31^-s − 24.5·33^-s + 14.2i·37^-s + 51.2·39^-s + ⋯

L(s) = 1

− 1.13i·3^-s − 0.254·7^-s − 0.283·9^-s − 0.656i·11^-s + 1.16i·13^-s + 0.601i·17^-s − 0.216·19^-s + 0.287i·21^-s − 0.965i·23^-s − 0.811i·27^-s + 0.807i·29^-s + (0.490 − 0.871i)31^-s − 0.744·33^-s + 0.384i·37^-s + 1.31·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$0.490 - 0.871i$
Analytic conductor:	$84.4688$
Root analytic conductor:	$9.19069$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3100} (1301, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3100,\ (\ :1),\ 0.490 - 0.871i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9222814813$
$L(\frac12)$	$\approx$	$0.9222814813$
$L(2)$		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 $
	31	$ 1 + (-15.2 + 27.0i)T $
good	3	$ 1 + 3.39iT - 9T^{2} $
	7	$ 1 + 1.77T + 49T^{2} $
	11	$ 1 + 7.22iT - 121T^{2} $
	13	$ 1 - 15.0iT - 169T^{2} $
	17	$ 1 - 10.2iT - 289T^{2} $
	19	$ 1 + 4.10T + 361T^{2} $
	23	$ 1 + 22.1iT - 529T^{2} $
	29	$ 1 - 23.4iT - 841T^{2} $
	37	$ 1 - 14.2iT - 1.36e3T^{2} $
	41	$ 1 + 59.1T + 1.68e3T^{2} $
	43	$ 1 - 45.7iT - 1.84e3T^{2} $
	47	$ 1 + 0.904T + 2.20e3T^{2} $
	53	$ 1 + 3.44iT - 2.80e3T^{2} $
	59	$ 1 + 95.8T + 3.48e3T^{2} $
	61	$ 1 - 28.0iT - 3.72e3T^{2} $
	67	$ 1 - 96.7T + 4.48e3T^{2} $
	71	$ 1 + 36.5T + 5.04e3T^{2} $
	73	$ 1 - 50.7iT - 5.32e3T^{2} $
	79	$ 1 - 102. iT - 6.24e3T^{2} $
	83	$ 1 - 49.3iT - 6.88e3T^{2} $
	89	$ 1 - 151. iT - 7.92e3T^{2} $
	97	$ 1 - 99.2T + 9.40e3T^{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

−8.360963525216550960265872169581, −8.044507138766550437972978019490, −6.87517684059694277759021527160, −6.62150972318794228135451282088, −5.91141073273564395945171426149, −4.76729827760745028922764751656, −3.92664404812100056356629983253, −2.84270228541881702335325300217, −1.89790029693091021324703146251, −1.05841808983669588416945051194, 0.21802576691164030083958202421, 1.72042776107068131977484411658, 3.02776655240731945811818498382, 3.59543701632050963084177466784, 4.62114206084887368729164225411, 5.11511224513997971969495524012, 5.96072497871761081907448087224, 6.96750903565874350969985471484, 7.66036840114693391131315473018, 8.520132722813356461638548721321

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9222814813\)
\(L(\frac12)\)	\(\approx\)	\(0.9222814813\)
\(L(2)\)		not available
\(L(1)\)		not available

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