LMFDB - L-function 2-50-5.4-c25-0-19

Dirichlet series

L(s) = 1

− 4.09e3i·2^-s + 1.75e6i·3^-s − 1.67e7·4^-s + 7.20e9·6^-s + 3.04e10i·7^-s + 6.87e10i·8^-s − 2.24e12·9^-s + 2.58e12·11^-s − 2.94e13i·12^-s − 9.57e13i·13^-s + 1.24e14·14^-s + 2.81e14·16^-s + 1.64e15i·17^-s + 9.18e15i·18^-s − 4.95e15·19^-s + ⋯

L(s) = 1

− 0.707i·2^-s + 1.90i·3^-s − 0.5·4^-s + 1.35·6^-s + 0.830i·7^-s + 0.353i·8^-s − 2.64·9^-s + 0.248·11^-s − 0.954i·12^-s − 1.13i·13^-s + 0.587·14^-s + 0.250·16^-s + 0.685i·17^-s + 1.87i·18^-s − 0.513·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(26-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$50$ = $2 \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$197.998$
Root analytic conductor:	$14.0711$
Motivic weight:	$25$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{50} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 50,\ (\ :25/2),\ 0.894 - 0.447i)$

Particular Values

$L(13)$	$\approx$	$1.502436214$
$L(\frac12)$	$\approx$	$1.502436214$
$L(\frac{27}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 + 4.09e3iT $
	5	$ 1 $
good	3	$ 1 - 1.75e6iT - 8.47e11T^{2} $
	7	$ 1 - 3.04e10iT - 1.34e21T^{2} $
	11	$ 1 - 2.58e12T + 1.08e26T^{2} $
	13	$ 1 + 9.57e13iT - 7.05e27T^{2} $
	17	$ 1 - 1.64e15iT - 5.77e30T^{2} $
	19	$ 1 + 4.95e15T + 9.30e31T^{2} $
	23	$ 1 + 1.07e16iT - 1.10e34T^{2} $
	29	$ 1 - 1.36e18T + 3.63e36T^{2} $
	31	$ 1 + 4.41e18T + 1.92e37T^{2} $
	37	$ 1 + 1.01e19iT - 1.60e39T^{2} $
	41	$ 1 - 1.58e20T + 2.08e40T^{2} $
	43	$ 1 - 1.83e20iT - 6.86e40T^{2} $
	47	$ 1 + 1.40e21iT - 6.34e41T^{2} $
	53	$ 1 + 1.99e21iT - 1.27e43T^{2} $
	59	$ 1 - 4.16e21T + 1.86e44T^{2} $
	61	$ 1 - 3.42e22T + 4.29e44T^{2} $
	67	$ 1 + 8.67e22iT - 4.48e45T^{2} $
	71	$ 1 + 5.13e22T + 1.91e46T^{2} $
	73	$ 1 - 3.49e22iT - 3.82e46T^{2} $
	79	$ 1 + 2.91e23T + 2.75e47T^{2} $
	83	$ 1 + 1.64e24iT - 9.48e47T^{2} $
	89	$ 1 + 8.74e23T + 5.42e48T^{2} $
	97	$ 1 + 1.00e25iT - 4.66e49T^{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

−10.71675471511308375567266006236, −10.04133770163744425313673609264, −9.023929604079708775958183051652, −8.313185665227651505057453113787, −5.90116215447072855422268688174, −5.13278555700569194572785574665, −4.05822314050636428806953043342, −3.23213051626936932778772431462, −2.25918726864773320779923178011, −0.42616984809084989254387528289, 0.64103250256461753962114471473, 1.43544321184850937272855687455, 2.59864172260909496266024169875, 4.16300452274081975773967046190, 5.67549939391788367499252574855, 6.76821926088595283169627886823, 7.16856956588130376106236261021, 8.193066857369770071121484767403, 9.249830944209383730876032893081, 11.07246334963077642666033925190

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 $\F_{q}$

\(L(13)\)	\(\approx\)	\(1.502436214\)
\(L(\frac12)\)	\(\approx\)	\(1.502436214\)
\(L(\frac{27}{2})\)		not available
\(L(1)\)		not available

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