LMFDB - L-function 2-5e2-5.4-c33-0-20

Dirichlet series

L(s) = 1

− 1.78e5i·2^-s + 8.55e7i·3^-s − 2.34e10·4^-s + 1.52e13·6^-s + 8.09e13i·7^-s + 2.64e15i·8^-s − 1.75e15·9^-s + 1.95e17·11^-s − 2.00e18i·12^-s − 2.59e18i·13^-s + 1.44e19·14^-s + 2.72e20·16^-s + 1.52e20i·17^-s + 3.13e20i·18^-s − 1.39e21·19^-s + ⋯

L(s) = 1

− 1.92i·2^-s + 1.14i·3^-s − 2.72·4^-s + 2.21·6^-s + 0.920i·7^-s + 3.32i·8^-s − 0.315·9^-s + 1.28·11^-s − 3.12i·12^-s − 1.08i·13^-s + 1.77·14^-s + 3.69·16^-s + 0.758i·17^-s + 0.608i·18^-s − 1.10·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$25$ = $5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$172.457$
Root analytic conductor:	$13.1322$
Motivic weight:	$33$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{25} (24, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 25,\ (\ :33/2),\ 0.447 + 0.894i)$

Particular Values

$L(17)$	$\approx$	$1.820403983$
$L(\frac12)$	$\approx$	$1.820403983$
$L(\frac{35}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	5	$ 1 $
good	2	$ 1 + 1.78e5iT - 8.58e9T^{2} $
	3	$ 1 - 8.55e7iT - 5.55e15T^{2} $
	7	$ 1 - 8.09e13iT - 7.73e27T^{2} $
	11	$ 1 - 1.95e17T + 2.32e34T^{2} $
	13	$ 1 + 2.59e18iT - 5.75e36T^{2} $
	17	$ 1 - 1.52e20iT - 4.02e40T^{2} $
	19	$ 1 + 1.39e21T + 1.58e42T^{2} $
	23	$ 1 + 3.40e22iT - 8.65e44T^{2} $
	29	$ 1 - 1.49e24T + 1.81e48T^{2} $
	31	$ 1 + 1.38e24T + 1.64e49T^{2} $
	37	$ 1 + 1.15e26iT - 5.63e51T^{2} $
	41	$ 1 + 6.35e25T + 1.66e53T^{2} $
	43	$ 1 + 2.64e26iT - 8.02e53T^{2} $
	47	$ 1 + 1.03e27iT - 1.51e55T^{2} $
	53	$ 1 - 5.20e28iT - 7.96e56T^{2} $
	59	$ 1 - 6.17e28T + 2.74e58T^{2} $
	61	$ 1 + 1.21e29T + 8.23e58T^{2} $
	67	$ 1 - 2.87e29iT - 1.82e60T^{2} $
	71	$ 1 + 2.84e30T + 1.23e61T^{2} $
	73	$ 1 + 2.67e30iT - 3.08e61T^{2} $
	79	$ 1 - 7.84e30T + 4.18e62T^{2} $
	83	$ 1 - 1.28e31iT - 2.13e63T^{2} $
	89	$ 1 - 1.56e32T + 2.13e64T^{2} $
	97	$ 1 + 4.84e32iT - 3.65e65T^{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

−10.82756889000433769920772541180, −10.28686112705090042727918921967, −9.126100293122948194051962237043, −8.574004083715004413833926071056, −5.81750708865092065861956507503, −4.56629432792764033702605081449, −3.89380041802622521165191720247, −2.87585005539811733181318198526, −1.84204087753400759778753592811, −0.61292131097700950164597956546, 0.62082918756665698136625786067, 1.49258555100176961134242711108, 3.84416119241801270133099970908, 4.73856239416812249182759773056, 6.39697606974394641979843596040, 6.75018016504118560731951358259, 7.57822188280177280552188596707, 8.672965953819221211431385927046, 9.763286711459023318367015771665, 11.88407585717827659476738825907

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 $\F_{q}$

\(L(17)\)	\(\approx\)	\(1.820403983\)
\(L(\frac12)\)	\(\approx\)	\(1.820403983\)
\(L(\frac{35}{2})\)		not available
\(L(1)\)		not available

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