LMFDB - L-function 2-3100-31.30-c2-0-17

Dirichlet series

L(s) = 1

− 5.24i·3^-s − 8.32·7^-s − 18.4·9^-s + 2.20i·11^-s − 13.9i·13^-s + 6.60i·17^-s − 18.1·19^-s + 43.6i·21^-s − 14.7i·23^-s + 49.7i·27^-s + 46.8i·29^-s + (28.0 + 13.1i)31^-s + 11.5·33^-s − 23.7i·37^-s − 73.2·39^-s + ⋯

L(s) = 1

− 1.74i·3^-s − 1.18·7^-s − 2.05·9^-s + 0.200i·11^-s − 1.07i·13^-s + 0.388i·17^-s − 0.954·19^-s + 2.07i·21^-s − 0.641i·23^-s + 1.84i·27^-s + 1.61i·29^-s + (0.904 + 0.425i)31^-s + 0.350·33^-s − 0.642i·37^-s − 1.87·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$0.904 + 0.425i$
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:	$(2,\ 3100,\ (\ :1),\ 0.904 + 0.425i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.8274544485$
$L(\frac12)$	$\approx$	$0.8274544485$
$L(2)$		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 + (-28.0 - 13.1i)T $
good	3	$ 1 + 5.24iT - 9T^{2} $
	7	$ 1 + 8.32T + 49T^{2} $
	11	$ 1 - 2.20iT - 121T^{2} $
	13	$ 1 + 13.9iT - 169T^{2} $
	17	$ 1 - 6.60iT - 289T^{2} $
	19	$ 1 + 18.1T + 361T^{2} $
	23	$ 1 + 14.7iT - 529T^{2} $
	29	$ 1 - 46.8iT - 841T^{2} $
	37	$ 1 + 23.7iT - 1.36e3T^{2} $
	41	$ 1 + 10.3T + 1.68e3T^{2} $
	43	$ 1 - 68.6iT - 1.84e3T^{2} $
	47	$ 1 + 22.3T + 2.20e3T^{2} $
	53	$ 1 - 16.9iT - 2.80e3T^{2} $
	59	$ 1 + 34.4T + 3.48e3T^{2} $
	61	$ 1 - 38.4iT - 3.72e3T^{2} $
	67	$ 1 - 96.6T + 4.48e3T^{2} $
	71	$ 1 + 89.2T + 5.04e3T^{2} $
	73	$ 1 - 28.5iT - 5.32e3T^{2} $
	79	$ 1 + 148. iT - 6.24e3T^{2} $
	83	$ 1 + 64.1iT - 6.88e3T^{2} $
	89	$ 1 + 57.9iT - 7.92e3T^{2} $
	97	$ 1 + 119.T + 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

−8.389889456809696488355145448522, −7.59868443809875906075013892357, −6.93411661223431049715590559137, −6.30434147904277883184425628041, −5.86820643851210919591389651604, −4.70683468373123668319126644828, −3.30117439367667438135125524967, −2.76134002879783648793900368250, −1.70268196332460589583741855925, −0.67435067773620791391126738906, 0.27156660575958979511697671425, 2.27346701569897962428632174563, 3.21010749294721554702938684413, 3.94960086988846823414657317554, 4.48587295863179325139258372912, 5.42162165832569107260267832342, 6.23053513090380013612994789917, 6.83532808382667335171134687777, 8.126498443438434500217243294966, 8.810583390948876033315999906614

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.8274544485\)
\(L(\frac12)\)	\(\approx\)	\(0.8274544485\)
\(L(2)\)		not available
\(L(1)\)		not available

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