LMFDB - L-function 2-525-5.4-c3-0-54

Dirichlet series

L(s) = 1

− 4.23i·2^-s − 3i·3^-s − 9.94·4^-s − 12.7·6^-s − 7i·7^-s + 8.23i·8^-s − 9·9^-s − 41.5·11^-s + 29.8i·12^-s − 88.9i·13^-s − 29.6·14^-s − 44.6·16^-s − 120. i·17^-s + 38.1i·18^-s + 112.·19^-s + ⋯

L(s) = 1

− 1.49i·2^-s − 0.577i·3^-s − 1.24·4^-s − 0.864·6^-s − 0.377i·7^-s + 0.363i·8^-s − 0.333·9^-s − 1.13·11^-s + 0.717i·12^-s − 1.89i·13^-s − 0.566·14^-s − 0.697·16^-s − 1.71i·17^-s + 0.499i·18^-s + 1.35·19^-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

$L(2)$	$\approx$	$0.9432985033$
$L(\frac12)$	$\approx$	$0.9432985033$
$L(\frac{5}{2})$		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 + 4.23iT - 8T^{2} $
	11	$ 1 + 41.5T + 1.33e3T^{2} $
	13	$ 1 + 88.9iT - 2.19e3T^{2} $
	17	$ 1 + 120. iT - 4.91e3T^{2} $
	19	$ 1 - 112.T + 6.85e3T^{2} $
	23	$ 1 - 115. iT - 1.21e4T^{2} $
	29	$ 1 - 144.T + 2.43e4T^{2} $
	31	$ 1 + 258.T + 2.97e4T^{2} $
	37	$ 1 - 48.3iT - 5.06e4T^{2} $
	41	$ 1 - 200.T + 6.89e4T^{2} $
	43	$ 1 - 218. iT - 7.95e4T^{2} $
	47	$ 1 - 575. iT - 1.03e5T^{2} $
	53	$ 1 - 184. iT - 1.48e5T^{2} $
	59	$ 1 - 151.T + 2.05e5T^{2} $
	61	$ 1 + 529.T + 2.26e5T^{2} $
	67	$ 1 - 1.28iT - 3.00e5T^{2} $
	71	$ 1 + 61.4T + 3.57e5T^{2} $
	73	$ 1 + 484. iT - 3.89e5T^{2} $
	79	$ 1 + 878.T + 4.93e5T^{2} $
	83	$ 1 + 491. iT - 5.71e5T^{2} $
	89	$ 1 - 415.T + 7.04e5T^{2} $
	97	$ 1 + 1.03e3iT - 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

−9.948035445078937996404182533189, −9.215197978436076251178106851999, −7.76712190599655738149455224128, −7.38996629216998087385113079209, −5.68003048887681946222963262919, −4.83976750361355594094720533691, −3.13253116104481272972588001508, −2.83528033427096463647666197159, −1.21100416868405288663222589763, −0.30721832406503930222481964623, 2.18498110468207537380224385567, 3.88900486469867854799395718820, 4.90853104921410795360070234082, 5.70485548349316412714677842163, 6.59731488510146135835771490082, 7.50474014685843443155313603229, 8.505025928829268358651699910681, 9.019983068034292439466931812387, 10.10178087538070155270015979860, 11.04569102467808904206876435311

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.9432985033\)
\(L(\frac12)\)	\(\approx\)	\(0.9432985033\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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