LMFDB - L-function 2-8280-69.68-c1-0-16

Dirichlet series

L(s) = 1

− 5^-s + 3.21i·7^-s − 4.83·11^-s − 1.11·13^-s + 5.43·17^-s − 1.24i·19^-s + (1.90 + 4.40i)23^-s + 25^-s − 2.73i·29^-s + 8.10·31^-s − 3.21i·35^-s + 7.85i·37^-s − 0.525i·41^-s + 1.68i·43^-s − 2.16i·47^-s + ⋯

L(s) = 1

− 0.447·5^-s + 1.21i·7^-s − 1.45·11^-s − 0.309·13^-s + 1.31·17^-s − 0.285i·19^-s + (0.396 + 0.918i)23^-s + 0.200·25^-s − 0.508i·29^-s + 1.45·31^-s − 0.542i·35^-s + 1.29i·37^-s − 0.0821i·41^-s + 0.257i·43^-s − 0.315i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8280$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$-0.853 - 0.520i$
Analytic conductor:	$66.1161$
Root analytic conductor:	$8.13118$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8280} (1241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8280,\ (\ :1/2),\ -0.853 - 0.520i)$

Particular Values

$L(1)$	$\approx$	$0.9563537499$
$L(\frac12)$	$\approx$	$0.9563537499$
$L(\frac{3}{2})$		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 $
	5	$ 1 + T $
	23	$ 1 + (-1.90 - 4.40i)T $
good	7	$ 1 - 3.21iT - 7T^{2} $
	11	$ 1 + 4.83T + 11T^{2} $
	13	$ 1 + 1.11T + 13T^{2} $
	17	$ 1 - 5.43T + 17T^{2} $
	19	$ 1 + 1.24iT - 19T^{2} $
	29	$ 1 + 2.73iT - 29T^{2} $
	31	$ 1 - 8.10T + 31T^{2} $
	37	$ 1 - 7.85iT - 37T^{2} $
	41	$ 1 + 0.525iT - 41T^{2} $
	43	$ 1 - 1.68iT - 43T^{2} $
	47	$ 1 + 2.16iT - 47T^{2} $
	53	$ 1 + 4.59T + 53T^{2} $
	59	$ 1 - 4.37iT - 59T^{2} $
	61	$ 1 + 11.3iT - 61T^{2} $
	67	$ 1 - 10.8iT - 67T^{2} $
	71	$ 1 - 1.30iT - 71T^{2} $
	73	$ 1 - 15.0T + 73T^{2} $
	79	$ 1 - 2.06iT - 79T^{2} $
	83	$ 1 - 2.56T + 83T^{2} $
	89	$ 1 - 4.37T + 89T^{2} $
	97	$ 1 - 14.8iT - 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

−8.001981324153346174410267953405, −7.66855449167481531287136005430, −6.69021849376229646480291115985, −5.92717122439990115717831940526, −5.16295876441573805248359062405, −4.92481491493148357781494116234, −3.67894582492214199240795884329, −2.86550127123167017830146816865, −2.41239330817489106767836277476, −1.10777707546808159323236646671, 0.26400579478204336343690292604, 1.12210425967831664116163979475, 2.44541211838262573674392687847, 3.19513649928995297771457488168, 3.94409593602400680822542016971, 4.75132988522272339392020489484, 5.27553816296501006893876777196, 6.19988902070916216083686563702, 7.00222993072098867610010296629, 7.69264994398050182045781435092

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9563537499\)
\(L(\frac12)\)	\(\approx\)	\(0.9563537499\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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