LMFDB - L-function 2-1183-13.12-c1-0-77

Dirichlet series

L(s) = 1

− 2.70i·2^-s − 0.345·3^-s − 5.30·4^-s − 3.25i·5^-s + 0.935i·6^-s − i·7^-s + 8.94i·8^-s − 2.88·9^-s − 8.80·10^-s − 1.84i·11^-s + 1.83·12^-s − 2.70·14^-s + 1.12i·15^-s + 13.5·16^-s − 2.15·17^-s + 7.78i·18^-s + ⋯

L(s) = 1

− 1.91i·2^-s − 0.199·3^-s − 2.65·4^-s − 1.45i·5^-s + 0.381i·6^-s − 0.377i·7^-s + 3.16i·8^-s − 0.960·9^-s − 2.78·10^-s − 0.556i·11^-s + 0.530·12^-s − 0.722·14^-s + 0.291i·15^-s + 3.38·16^-s − 0.522·17^-s + 1.83i·18^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1183$ = $7 \cdot 13^{2}$
Sign:	$0.960 - 0.277i$
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:	$(2,\ 1183,\ (\ :1/2),\ 0.960 - 0.277i)$

Particular Values

$L(1)$	$\approx$	$0.3327579879$
$L(\frac12)$	$\approx$	$0.3327579879$
$L(\frac{3}{2})$		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 + 2.70iT - 2T^{2} $
	3	$ 1 + 0.345T + 3T^{2} $
	5	$ 1 + 3.25iT - 5T^{2} $
	11	$ 1 + 1.84iT - 11T^{2} $
	17	$ 1 + 2.15T + 17T^{2} $
	19	$ 1 - 2.40iT - 19T^{2} $
	23	$ 1 + 1.81T + 23T^{2} $
	29	$ 1 + 2.73T + 29T^{2} $
	31	$ 1 + 1.74iT - 31T^{2} $
	37	$ 1 + 5.93iT - 37T^{2} $
	41	$ 1 - 4.22iT - 41T^{2} $
	43	$ 1 - 8.68T + 43T^{2} $
	47	$ 1 - 5.87iT - 47T^{2} $
	53	$ 1 + 9.30T + 53T^{2} $
	59	$ 1 + 10.7iT - 59T^{2} $
	61	$ 1 - 10.1T + 61T^{2} $
	67	$ 1 - 0.826iT - 67T^{2} $
	71	$ 1 - 2.35iT - 71T^{2} $
	73	$ 1 - 3.19iT - 73T^{2} $
	79	$ 1 - 0.801T + 79T^{2} $
	83	$ 1 + 9.97iT - 83T^{2} $
	89	$ 1 - 15.1iT - 89T^{2} $
	97	$ 1 + 9.23iT - 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

−9.232079132563024585592615036860, −8.521174371210407992429156271166, −7.943363197486312755445399361530, −5.96131201463439674701678599961, −5.19802169955536283199425621691, −4.36405106386360844549463796867, −3.57720711204680748562963281446, −2.37205340676392858162588763742, −1.18965994910072296294388742785, −0.16605927014247876806878233182, 2.65063975360568580529352310621, 3.84887075221206311240758495006, 4.98108629902582863248233027233, 5.80611887539142500927832449512, 6.50195376301546564908775790292, 7.05681628253771572088796377984, 7.83381297695161463800303041413, 8.691848200566319155149927165536, 9.408506021176553613300040095897, 10.32599671922788479226898292045

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3327579879\)
\(L(\frac12)\)	\(\approx\)	\(0.3327579879\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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