LMFDB - L-function 2-1690-65.64-c1-0-16

Dirichlet series

L(s) = 1

− 2^-s − 0.956i·3^-s + 4^-s + (−2.19 − 0.426i)5^-s + 0.956i·6^-s − 2.15·7^-s − 8^-s + 2.08·9^-s + (2.19 + 0.426i)10^-s − 1.30i·11^-s − 0.956i·12^-s + 2.15·14^-s + (−0.408 + 2.09i)15^-s + 16^-s + 2.87i·17^-s − 2.08·18^-s + ⋯

L(s) = 1

− 0.707·2^-s − 0.552i·3^-s + 0.5·4^-s + (−0.981 − 0.190i)5^-s + 0.390i·6^-s − 0.813·7^-s − 0.353·8^-s + 0.695·9^-s + (0.694 + 0.134i)10^-s − 0.393i·11^-s − 0.276i·12^-s + 0.574·14^-s + (−0.105 + 0.542i)15^-s + 0.250·16^-s + 0.697i·17^-s − 0.491·18^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1690$ = $2 \cdot 5 \cdot 13^{2}$
Sign:	$0.986 + 0.160i$
Analytic conductor:	$13.4947$
Root analytic conductor:	$3.67351$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1690} (1689, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1690,\ (\ :1/2),\ 0.986 + 0.160i)$

Particular Values

$L(1)$	$\approx$	$0.8111412388$
$L(\frac12)$	$\approx$	$0.8111412388$
$L(\frac{3}{2})$		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 + T $
	5	$ 1 + (2.19 + 0.426i)T $
	13	$ 1 $
good	3	$ 1 + 0.956iT - 3T^{2} $
	7	$ 1 + 2.15T + 7T^{2} $
	11	$ 1 + 1.30iT - 11T^{2} $
	17	$ 1 - 2.87iT - 17T^{2} $
	19	$ 1 - 2.65iT - 19T^{2} $
	23	$ 1 - 5.11iT - 23T^{2} $
	29	$ 1 + 7.96T + 29T^{2} $
	31	$ 1 + 4.12iT - 31T^{2} $
	37	$ 1 - 2.98T + 37T^{2} $
	41	$ 1 - 8.69iT - 41T^{2} $
	43	$ 1 - 3.35iT - 43T^{2} $
	47	$ 1 - 7.30T + 47T^{2} $
	53	$ 1 + 10.0iT - 53T^{2} $
	59	$ 1 + 14.6iT - 59T^{2} $
	61	$ 1 - 11.2T + 61T^{2} $
	67	$ 1 - 11.0T + 67T^{2} $
	71	$ 1 - 14.0iT - 71T^{2} $
	73	$ 1 - 3.83T + 73T^{2} $
	79	$ 1 - 8.94T + 79T^{2} $
	83	$ 1 - 13.9T + 83T^{2} $
	89	$ 1 - 15.6iT - 89T^{2} $
	97	$ 1 + 5.05T + 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

−9.545630024806744282534477335016, −8.187160266838291923423183769500, −7.966328717829541115271978769766, −7.04241387814099402890942648699, −6.42687639965473466559250771549, −5.43630882601796477987972784612, −4.00767588373361771749929165552, −3.41585230483274317690351108324, −1.98156564198715293156783432553, −0.77197306347038137606681935431, 0.60565249501384645446529167344, 2.36862162277753188001805688383, 3.44894992395306286143939244582, 4.21337206007590584618058177927, 5.17132510841250088708139463546, 6.46680456367479088454850813227, 7.18955244169087614666417994436, 7.62308081340457130412795653597, 8.896229777913968784752284890559, 9.213930509149533577516695340170

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.8111412388\)
\(L(\frac12)\)	\(\approx\)	\(0.8111412388\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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