LMFDB - L-function 2-1183-13.12-c1-0-37

Dirichlet series

L(s) = 1

− 2.63i·2^-s − 2.71·3^-s − 4.95·4^-s + 2.08i·5^-s + 7.15i·6^-s − i·7^-s + 7.80i·8^-s + 4.34·9^-s + 5.49·10^-s − 3.60i·11^-s + 13.4·12^-s − 2.63·14^-s − 5.64i·15^-s + 10.6·16^-s + 5.76·17^-s − 11.4i·18^-s + ⋯

L(s) = 1

− 1.86i·2^-s − 1.56·3^-s − 2.47·4^-s + 0.931i·5^-s + 2.91i·6^-s − 0.377i·7^-s + 2.75i·8^-s + 1.44·9^-s + 1.73·10^-s − 1.08i·11^-s + 3.87·12^-s − 0.704·14^-s − 1.45i·15^-s + 2.66·16^-s + 1.39·17^-s − 2.70i·18^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1183$ = $7 \cdot 13^{2}$
Sign:	$-0.722 - 0.691i$
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:	$(2,\ 1183,\ (\ :1/2),\ -0.722 - 0.691i)$

Particular Values

$L(1)$	$\approx$	$0.4834545633$
$L(\frac12)$	$\approx$	$0.4834545633$
$L(\frac{3}{2})$		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 + 2.63iT - 2T^{2} $
	3	$ 1 + 2.71T + 3T^{2} $
	5	$ 1 - 2.08iT - 5T^{2} $
	11	$ 1 + 3.60iT - 11T^{2} $
	17	$ 1 - 5.76T + 17T^{2} $
	19	$ 1 + 1.36iT - 19T^{2} $
	23	$ 1 - 4.22T + 23T^{2} $
	29	$ 1 + 8.29T + 29T^{2} $
	31	$ 1 - 0.734iT - 31T^{2} $
	37	$ 1 - 2.16iT - 37T^{2} $
	41	$ 1 - 2.86iT - 41T^{2} $
	43	$ 1 + 4.26T + 43T^{2} $
	47	$ 1 + 8.68iT - 47T^{2} $
	53	$ 1 + 10.9T + 53T^{2} $
	59	$ 1 - 6.75iT - 59T^{2} $
	61	$ 1 - 10.5T + 61T^{2} $
	67	$ 1 + 4.70iT - 67T^{2} $
	71	$ 1 + 13.3iT - 71T^{2} $
	73	$ 1 + 4.36iT - 73T^{2} $
	79	$ 1 + 1.73T + 79T^{2} $
	83	$ 1 + 5.74iT - 83T^{2} $
	89	$ 1 + 8.41iT - 89T^{2} $
	97	$ 1 + 6.20iT - 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

−9.863079995018046003575123205275, −8.841019390872656555958665758098, −7.62509815350510131743096850654, −6.53548558069615650846756318624, −5.55741999680987932167974163233, −4.87486130499712806675426915863, −3.63356689871966636047051305499, −3.02117415719902330733054910537, −1.42160356070551054379468305621, −0.33750611151436626308760498061, 1.13486452745248995933080848243, 3.97973639819539196554029603237, 5.00812920076772280876542131504, 5.25925178879715944619764011584, 5.98056281629152183196283501996, 6.86891302194130329070180747521, 7.53687856611492780010305173950, 8.384298876372465558319238644563, 9.415992430268304349604157158072, 9.842625080129170932514170340626

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4834545633\)
\(L(\frac12)\)	\(\approx\)	\(0.4834545633\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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