LMFDB - L-function 2-1183-13.12-c1-0-4

Dirichlet series

L(s) = 1

− 2.22i·2^-s + 0.549·3^-s − 2.92·4^-s + 4.22i·5^-s − 1.22i·6^-s − i·7^-s + 2.06i·8^-s − 2.69·9^-s + 9.36·10^-s − 0.549i·11^-s − 1.60·12^-s − 2.22·14^-s + 2.31i·15^-s − 1.28·16^-s + 2.37·17^-s + 5.98i·18^-s + ⋯

L(s) = 1

− 1.56i·2^-s + 0.317·3^-s − 1.46·4^-s + 1.88i·5^-s − 0.498i·6^-s − 0.377i·7^-s + 0.728i·8^-s − 0.899·9^-s + 2.96·10^-s − 0.165i·11^-s − 0.464·12^-s − 0.593·14^-s + 0.598i·15^-s − 0.320·16^-s + 0.576·17^-s + 1.41i·18^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1183$ = $7 \cdot 13^{2}$
Sign:	$0.554 - 0.832i$
Analytic conductor:	$9.44630$
Root analytic conductor:	$3.07348$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1183} (337, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1183,\ (\ :1/2),\ 0.554 - 0.832i)$

Particular Values

$L(1)$	$\approx$	$0.6891263636$
$L(\frac12)$	$\approx$	$0.6891263636$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	7	$ 1 + iT $
	13	$ 1 $
good	2	$ 1 + 2.22iT - 2T^{2} $
	3	$ 1 - 0.549T + 3T^{2} $
	5	$ 1 - 4.22iT - 5T^{2} $
	11	$ 1 + 0.549iT - 11T^{2} $
	17	$ 1 - 2.37T + 17T^{2} $
	19	$ 1 - 3.61iT - 19T^{2} $
	23	$ 1 + 5.81T + 23T^{2} $
	29	$ 1 + 3.59T + 29T^{2} $
	31	$ 1 - 5.14iT - 31T^{2} $
	37	$ 1 + 0.329iT - 37T^{2} $
	41	$ 1 - 6.29iT - 41T^{2} $
	43	$ 1 + 3.22T + 43T^{2} $
	47	$ 1 - 8.20iT - 47T^{2} $
	53	$ 1 - 2.65T + 53T^{2} $
	59	$ 1 + 1.80iT - 59T^{2} $
	61	$ 1 - 0.609T + 61T^{2} $
	67	$ 1 + 10.3iT - 67T^{2} $
	71	$ 1 - 11.1iT - 71T^{2} $
	73	$ 1 - 4.90iT - 73T^{2} $
	79	$ 1 + 14.0T + 79T^{2} $
	83	$ 1 - 5.73iT - 83T^{2} $
	89	$ 1 + 7.46iT - 89T^{2} $
	97	$ 1 - 6.84iT - 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

−10.09561198270610859367309771999, −9.595339848450707508653893009574, −8.358932539305314319983662478075, −7.56865041467491066494608536648, −6.55063514696501701817312474353, −5.68684983047049584526945624676, −4.06603378116059055414201340466, −3.33189292157050453225091944178, −2.80031010405854466933858165534, −1.77481783129762205720600333027, 0.26772572594525316824304681138, 2.10858386739300633156347135544, 3.90088120398658949254749553004, 4.85571104418603014938920748418, 5.54898181080703554846974630671, 6.02403088277351423273605615847, 7.36855432376593242514391273212, 8.077393745146981867233621790219, 8.676925204716452761292030533410, 9.097445927867216656783678059541

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6891263636\)
\(L(\frac12)\)	\(\approx\)	\(0.6891263636\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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