LMFDB - L-function 2-525-5.4-c3-0-13

Dirichlet series

L(s) = 1

− 5i·2^-s − 3i·3^-s − 17·4^-s − 15·6^-s − 7i·7^-s + 45i·8^-s − 9·9^-s + 12·11^-s + 51i·12^-s + 30i·13^-s − 35·14^-s + 89·16^-s + 134i·17^-s + 45i·18^-s + 92·19^-s + ⋯

L(s) = 1

− 1.76i·2^-s − 0.577i·3^-s − 2.12·4^-s − 1.02·6^-s − 0.377i·7^-s + 1.98i·8^-s − 0.333·9^-s + 0.328·11^-s + 1.22i·12^-s + 0.640i·13^-s − 0.668·14^-s + 1.39·16^-s + 1.91i·17^-s + 0.589i·18^-s + 1.11·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$525$ = $3 \cdot 5^{2} \cdot 7$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$30.9760$
Root analytic conductor:	$5.56560$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{525} (274, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 525,\ (\ :3/2),\ 0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.166728381$
$L(\frac12)$	$\approx$	$1.166728381$
$L(\frac{5}{2})$		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 $
	7	$ 1 + 7iT $
good	2	$ 1 + 5iT - 8T^{2} $
	11	$ 1 - 12T + 1.33e3T^{2} $
	13	$ 1 - 30iT - 2.19e3T^{2} $
	17	$ 1 - 134iT - 4.91e3T^{2} $
	19	$ 1 - 92T + 6.85e3T^{2} $
	23	$ 1 - 112iT - 1.21e4T^{2} $
	29	$ 1 - 58T + 2.43e4T^{2} $
	31	$ 1 + 224T + 2.97e4T^{2} $
	37	$ 1 - 146iT - 5.06e4T^{2} $
	41	$ 1 - 18T + 6.89e4T^{2} $
	43	$ 1 - 340iT - 7.95e4T^{2} $
	47	$ 1 + 208iT - 1.03e5T^{2} $
	53	$ 1 + 754iT - 1.48e5T^{2} $
	59	$ 1 + 380T + 2.05e5T^{2} $
	61	$ 1 - 718T + 2.26e5T^{2} $
	67	$ 1 + 412iT - 3.00e5T^{2} $
	71	$ 1 + 960T + 3.57e5T^{2} $
	73	$ 1 - 1.06e3iT - 3.89e5T^{2} $
	79	$ 1 + 896T + 4.93e5T^{2} $
	83	$ 1 - 436iT - 5.71e5T^{2} $
	89	$ 1 - 1.03e3T + 7.04e5T^{2} $
	97	$ 1 - 702iT - 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

−10.42595755866739323090356494508, −9.674178732454685000049996961408, −8.809394589780622704061818389405, −7.85908873739547548724140952246, −6.63258564425627448251898906408, −5.34578897225955341319423305355, −4.04907078036679444790608634026, −3.32461736289031074119610053593, −1.89776166478418392284370755303, −1.18480633670555135579211835896, 0.40926267412199916161492296753, 2.97792479975793590736842334955, 4.38375148537717703945141083609, 5.24910831615715771093284791001, 5.88921960399178111643206010912, 7.08702854628003906995801771606, 7.66615696543583308656028630782, 8.887350438553412334698872153535, 9.227145245004230714359924725357, 10.25933895455305968310546091819

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.166728381\)
\(L(\frac12)\)	\(\approx\)	\(1.166728381\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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