LMFDB - L-function 2-338-13.12-c7-0-84

Dirichlet series

L(s) = 1

− 8i·2^-s + 12·3^-s − 64·4^-s − 210i·5^-s − 96i·6^-s − 1.01e3i·7^-s + 512i·8^-s − 2.04e3·9^-s − 1.68e3·10^-s − 1.09e3i·11^-s − 768·12^-s − 8.12e3·14^-s − 2.52e3i·15^-s + 4.09e3·16^-s − 1.47e4·17^-s + 1.63e4i·18^-s + ⋯

L(s) = 1

− 0.707i·2^-s + 0.256·3^-s − 0.5·4^-s − 0.751i·5^-s − 0.181i·6^-s − 1.11i·7^-s + 0.353i·8^-s − 0.934·9^-s − 0.531·10^-s − 0.247i·11^-s − 0.128·12^-s − 0.791·14^-s − 0.192i·15^-s + 0.250·16^-s − 0.725·17^-s + 0.660i·18^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$338$ = $2 \cdot 13^{2}$
Sign:	$0.554 - 0.832i$
Analytic conductor:	$105.586$
Root analytic conductor:	$10.2755$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{338} (337, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 338,\ (\ :7/2),\ 0.554 - 0.832i)$

Particular Values

$L(4)$	$\approx$	$0.1997245180$
$L(\frac12)$	$\approx$	$0.1997245180$
$L(\frac{9}{2})$		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 + 8iT $
	13	$ 1 $
good	3	$ 1 - 12T + 2.18e3T^{2} $
	5	$ 1 + 210iT - 7.81e4T^{2} $
	7	$ 1 + 1.01e3iT - 8.23e5T^{2} $
	11	$ 1 + 1.09e3iT - 1.94e7T^{2} $
	17	$ 1 + 1.47e4T + 4.10e8T^{2} $
	19	$ 1 + 3.99e4iT - 8.93e8T^{2} $
	23	$ 1 + 6.87e4T + 3.40e9T^{2} $
	29	$ 1 + 1.02e5T + 1.72e10T^{2} $
	31	$ 1 - 2.27e5iT - 2.75e10T^{2} $
	37	$ 1 + 1.60e5iT - 9.49e10T^{2} $
	41	$ 1 - 1.08e4iT - 1.94e11T^{2} $
	43	$ 1 - 6.30e5T + 2.71e11T^{2} $
	47	$ 1 + 4.72e5iT - 5.06e11T^{2} $
	53	$ 1 + 1.49e6T + 1.17e12T^{2} $
	59	$ 1 + 2.64e6iT - 2.48e12T^{2} $
	61	$ 1 - 8.27e5T + 3.14e12T^{2} $
	67	$ 1 + 1.26e5iT - 6.06e12T^{2} $
	71	$ 1 + 1.41e6iT - 9.09e12T^{2} $
	73	$ 1 + 9.80e5iT - 1.10e13T^{2} $
	79	$ 1 + 3.56e6T + 1.92e13T^{2} $
	83	$ 1 - 5.67e6iT - 2.71e13T^{2} $
	89	$ 1 - 1.19e7iT - 4.42e13T^{2} $
	97	$ 1 - 8.68e6iT - 8.07e13T^{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

−9.466361913990726472536375141835, −8.798923823847885865299870550200, −7.925011510311354762228810761728, −6.70590073922238715424041614328, −5.32675636586619712859858927556, −4.38277525488837341635613550839, −3.38471178110319624090115645779, −2.17418299279376144401497916770, −0.857719501888038576149720104183, −0.04922624623075864851297871490, 2.01833355020686133479772528893, 2.98459595240113472773452060380, 4.24284580436962355296782159004, 5.76565046640810504681242304186, 6.08210471049016554855184256481, 7.42121979422909585781242846905, 8.277609522282767611694104794148, 9.075672135532862336299556775536, 9.994092214543195973380672177484, 11.16101331040623644207714090406

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(0.1997245180\)
\(L(\frac12)\)	\(\approx\)	\(0.1997245180\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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