LMFDB - L-function 2-2925-5.4-c1-0-89

Dirichlet series

L(s) = 1

− 2.41i·2^-s − 3.82·4^-s − 4.82i·7^-s + 4.41i·8^-s − 3.41·11^-s − i·13^-s − 11.6·14^-s + 2.99·16^-s + 0.828i·17^-s − 0.585·19^-s + 8.24i·22^-s − 1.41i·23^-s − 2.41·26^-s + 18.4i·28^-s − 5.65·29^-s + ⋯

L(s) = 1

− 1.70i·2^-s − 1.91·4^-s − 1.82i·7^-s + 1.56i·8^-s − 1.02·11^-s − 0.277i·13^-s − 3.11·14^-s + 0.749·16^-s + 0.200i·17^-s − 0.134·19^-s + 1.75i·22^-s − 0.294i·23^-s − 0.473·26^-s + 3.49i·28^-s − 1.05·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2925$ = $3^{2} \cdot 5^{2} \cdot 13$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$23.3562$
Root analytic conductor:	$4.83282$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2925} (2224, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2925,\ (\ :1/2),\ 0.894 - 0.447i)$

Particular Values

$L(1)$	$\approx$	$0.3482334367$
$L(\frac12)$	$\approx$	$0.3482334367$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
	13	$ 1 + iT $
good	2	$ 1 + 2.41iT - 2T^{2} $
	7	$ 1 + 4.82iT - 7T^{2} $
	11	$ 1 + 3.41T + 11T^{2} $
	17	$ 1 - 0.828iT - 17T^{2} $
	19	$ 1 + 0.585T + 19T^{2} $
	23	$ 1 + 1.41iT - 23T^{2} $
	29	$ 1 + 5.65T + 29T^{2} $
	31	$ 1 - 1.75T + 31T^{2} $
	37	$ 1 - 8.48iT - 37T^{2} $
	41	$ 1 - 3.17T + 41T^{2} $
	43	$ 1 + 11.0iT - 43T^{2} $
	47	$ 1 + 4.82iT - 47T^{2} $
	53	$ 1 + 2.48iT - 53T^{2} $
	59	$ 1 - 1.75T + 59T^{2} $
	61	$ 1 + 8T + 61T^{2} $
	67	$ 1 - 2iT - 67T^{2} $
	71	$ 1 + 11.8T + 71T^{2} $
	73	$ 1 - 8.48iT - 73T^{2} $
	79	$ 1 - 8.48T + 79T^{2} $
	83	$ 1 - 3.17iT - 83T^{2} $
	89	$ 1 - 6T + 89T^{2} $
	97	$ 1 - 7.65iT - 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

−8.172294389468884886044883633176, −7.49650927844277435985215191707, −6.65447427738839426051983747430, −5.35086408500991917908583096351, −4.54814129428056655775229285992, −3.85372944223762213692159515795, −3.17732553418852512737806657407, −2.15061489877227527466998816280, −1.07481637229153370432379913790, −0.12222083067455853538549150965, 2.08794382190561377713461790859, 3.05826945255931175800205312836, 4.47245478180240232235890475515, 5.14240425184575759326624624004, 5.88917254452359214267892383475, 6.14004013680975663189727613131, 7.31969057837638895788842891122, 7.82869359105809546178662645937, 8.540688593338095718725537628900, 9.210433232528545517996952435293

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3482334367\)
\(L(\frac12)\)	\(\approx\)	\(0.3482334367\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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