LMFDB - L-function 2-84e2-12.11-c1-0-59

Dirichlet series

L(s) = 1

− 3.19i·5^-s − 1.57·11^-s + 0.670·13^-s + 5.91i·17^-s + 3.28i·19^-s + 3.95·23^-s − 5.18·25^-s − 0.585i·29^-s − 9.54i·31^-s + 7.69·37^-s − 3.77i·41^-s − 12.3i·43^-s − 7.34·47^-s + 13.3i·53^-s + 5.02i·55^-s + ⋯

L(s) = 1

− 1.42i·5^-s − 0.474·11^-s + 0.185·13^-s + 1.43i·17^-s + 0.754i·19^-s + 0.824·23^-s − 1.03·25^-s − 0.108i·29^-s − 1.71i·31^-s + 1.26·37^-s − 0.589i·41^-s − 1.88i·43^-s − 1.07·47^-s + 1.83i·53^-s + 0.677i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$-0.418 + 0.908i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (4607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ -0.418 + 0.908i)$

Particular Values

$L(1)$	$\approx$	$1.541225678$
$L(\frac12)$	$\approx$	$1.541225678$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 3.19iT - 5T^{2} $
	11	$ 1 + 1.57T + 11T^{2} $
	13	$ 1 - 0.670T + 13T^{2} $
	17	$ 1 - 5.91iT - 17T^{2} $
	19	$ 1 - 3.28iT - 19T^{2} $
	23	$ 1 - 3.95T + 23T^{2} $
	29	$ 1 + 0.585iT - 29T^{2} $
	31	$ 1 + 9.54iT - 31T^{2} $
	37	$ 1 - 7.69T + 37T^{2} $
	41	$ 1 + 3.77iT - 41T^{2} $
	43	$ 1 + 12.3iT - 43T^{2} $
	47	$ 1 + 7.34T + 47T^{2} $
	53	$ 1 - 13.3iT - 53T^{2} $
	59	$ 1 + 4.96T + 59T^{2} $
	61	$ 1 - 6.51T + 61T^{2} $
	67	$ 1 + 5.51iT - 67T^{2} $
	71	$ 1 - 6.85T + 71T^{2} $
	73	$ 1 - 5.84T + 73T^{2} $
	79	$ 1 + 6.08iT - 79T^{2} $
	83	$ 1 + 9.48T + 83T^{2} $
	89	$ 1 + 14.8iT - 89T^{2} $
	97	$ 1 + 0.183T + 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

−7.947066175399771679568623573175, −7.14618004941424701656082067507, −5.94000308004644991318613808881, −5.79057564256886055571737382551, −4.79545794937039645485467207976, −4.22372390535163963370909402864, −3.47961888210408122581356041342, −2.24468906808290068677783301947, −1.43185060418697309001650200817, −0.42415251724052350337304705359, 1.03597202484057456039076938345, 2.46082083703079318611540975559, 2.90583824248661985723622863901, 3.55648842100689838068123837739, 4.80700786580095348439690588153, 5.17652895719362160157591428894, 6.40775383098575705580645651744, 6.69485984682554250425018808698, 7.36741646178820913408976177093, 8.000154655658086150827316150808

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.541225678\)
\(L(\frac12)\)	\(\approx\)	\(1.541225678\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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