LMFDB - L-function 2-825-5.4-c3-0-90

Dirichlet series

L(s) = 1

− 4.59i·2^-s − 3i·3^-s − 13.1·4^-s − 13.7·6^-s − 20.6i·7^-s + 23.4i·8^-s − 9·9^-s + 11·11^-s + 39.3i·12^-s − 15.6i·13^-s − 94.8·14^-s + 3.04·16^-s − 72.9i·17^-s + 41.3i·18^-s − 61.0·19^-s + ⋯

L(s) = 1

− 1.62i·2^-s − 0.577i·3^-s − 1.63·4^-s − 0.937·6^-s − 1.11i·7^-s + 1.03i·8^-s − 0.333·9^-s + 0.301·11^-s + 0.946i·12^-s − 0.334i·13^-s − 1.81·14^-s + 0.0475·16^-s − 1.04i·17^-s + 0.541i·18^-s − 0.737·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$825$ = $3 \cdot 5^{2} \cdot 11$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$48.6765$
Root analytic conductor:	$6.97686$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{825} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 825,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$0.6907802788$
$L(\frac12)$	$\approx$	$0.6907802788$
$L(\frac{5}{2})$		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 + 3iT $
	5	$ 1 $
	11	$ 1 - 11T $
good	2	$ 1 + 4.59iT - 8T^{2} $
	7	$ 1 + 20.6iT - 343T^{2} $
	13	$ 1 + 15.6iT - 2.19e3T^{2} $
	17	$ 1 + 72.9iT - 4.91e3T^{2} $
	19	$ 1 + 61.0T + 6.85e3T^{2} $
	23	$ 1 + 13.6iT - 1.21e4T^{2} $
	29	$ 1 - 31.4T + 2.43e4T^{2} $
	31	$ 1 + 243.T + 2.97e4T^{2} $
	37	$ 1 - 65.4iT - 5.06e4T^{2} $
	41	$ 1 + 109.T + 6.89e4T^{2} $
	43	$ 1 + 121. iT - 7.95e4T^{2} $
	47	$ 1 - 519. iT - 1.03e5T^{2} $
	53	$ 1 + 542. iT - 1.48e5T^{2} $
	59	$ 1 + 109.T + 2.05e5T^{2} $
	61	$ 1 + 89.6T + 2.26e5T^{2} $
	67	$ 1 + 488. iT - 3.00e5T^{2} $
	71	$ 1 - 837.T + 3.57e5T^{2} $
	73	$ 1 - 351. iT - 3.89e5T^{2} $
	79	$ 1 - 831.T + 4.93e5T^{2} $
	83	$ 1 - 1.38e3iT - 5.71e5T^{2} $
	89	$ 1 + 1.52e3T + 7.04e5T^{2} $
	97	$ 1 - 426. iT - 9.12e5T^{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

−9.381209417171100281909254186867, −8.405866062174287063184659033401, −7.36522327953380782159788293864, −6.57187941560444325606935071834, −5.11699732600092866158281325013, −4.11179776452718045006170685983, −3.26201579935042730506599431038, −2.16793195719017385793928041165, −1.09216001699824977813326038165, −0.20107255527993311670372156453, 2.09999517394450833272101786499, 3.70537328102698842095350363597, 4.68722903454362901628490053571, 5.62085521939384431708198142679, 6.15291805912711404746796694026, 7.07966807477850504624411387769, 8.143487478487197967164420326605, 8.813085466762099566440708108838, 9.274929575743914524072385235985, 10.40362070462061816221611824122

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.6907802788\)
\(L(\frac12)\)	\(\approx\)	\(0.6907802788\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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