LMFDB - L-function 2-75-5.4-c21-0-15

Dirichlet series

L(s) = 1

− 2.84e3i·2^-s + 5.90e4i·3^-s − 5.99e6·4^-s + 1.67e8·6^-s + 3.63e8i·7^-s + 1.10e10i·8^-s − 3.48e9·9^-s + 1.45e10·11^-s − 3.53e11i·12^-s − 1.13e11i·13^-s + 1.03e12·14^-s + 1.89e13·16^-s − 8.58e12i·17^-s + 9.91e12i·18^-s + 2.92e13·19^-s + ⋯

L(s) = 1

− 1.96i·2^-s + 0.577i·3^-s − 2.85·4^-s + 1.13·6^-s + 0.486i·7^-s + 3.64i·8^-s − 0.333·9^-s + 0.169·11^-s − 1.64i·12^-s − 0.228i·13^-s + 0.954·14^-s + 4.30·16^-s − 1.03i·17^-s + 0.654i·18^-s + 1.09·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(22-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$75$ = $3 \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$209.608$
Root analytic conductor:	$14.4778$
Motivic weight:	$21$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{75} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 75,\ (\ :21/2),\ 0.894 + 0.447i)$

Particular Values

$L(11)$	$\approx$	$1.109382968$
$L(\frac12)$	$\approx$	$1.109382968$
$L(\frac{23}{2})$		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 - 5.90e4iT $
	5	$ 1 $
good	2	$ 1 + 2.84e3iT - 2.09e6T^{2} $
	7	$ 1 - 3.63e8iT - 5.58e17T^{2} $
	11	$ 1 - 1.45e10T + 7.40e21T^{2} $
	13	$ 1 + 1.13e11iT - 2.47e23T^{2} $
	17	$ 1 + 8.58e12iT - 6.90e25T^{2} $
	19	$ 1 - 2.92e13T + 7.14e26T^{2} $
	23	$ 1 - 1.55e14iT - 3.94e28T^{2} $
	29	$ 1 + 2.40e15T + 5.13e30T^{2} $
	31	$ 1 - 2.23e15T + 2.08e31T^{2} $
	37	$ 1 + 3.07e16iT - 8.55e32T^{2} $
	41	$ 1 + 1.03e17T + 7.38e33T^{2} $
	43	$ 1 - 1.65e17iT - 2.00e34T^{2} $
	47	$ 1 + 6.65e16iT - 1.30e35T^{2} $
	53	$ 1 + 4.35e17iT - 1.62e36T^{2} $
	59	$ 1 + 5.53e18T + 1.54e37T^{2} $
	61	$ 1 + 7.17e18T + 3.10e37T^{2} $
	67	$ 1 + 1.57e19iT - 2.22e38T^{2} $
	71	$ 1 - 2.64e19T + 7.52e38T^{2} $
	73	$ 1 + 1.34e19iT - 1.34e39T^{2} $
	79	$ 1 - 1.68e19T + 7.08e39T^{2} $
	83	$ 1 - 1.70e20iT - 1.99e40T^{2} $
	89	$ 1 - 3.12e20T + 8.65e40T^{2} $
	97	$ 1 - 9.49e20iT - 5.27e41T^{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

−10.70395271595365656400159869689, −9.540105257839947541989109318210, −9.213803361351557768225144693957, −7.85707229564663456303443403208, −5.55083429288188830203847058749, −4.78787285958715234041296435553, −3.59590640501014609903974231089, −2.90913331024900715429975787473, −1.83237557819244857294713399103, −0.72897532887133491439856541628, 0.29829578101666617833321259049, 1.35881730184474954745301644058, 3.49597279815061322255561467576, 4.57260614924339785961083986405, 5.67961360659597809460663018095, 6.55895443587506111554054254622, 7.36437350569723422554403866589, 8.203183620132552849886121851833, 9.136884268012323542494651105478, 10.31790327359193048529385548741

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 $\F_{q}$

\(L(11)\)	\(\approx\)	\(1.109382968\)
\(L(\frac12)\)	\(\approx\)	\(1.109382968\)
\(L(\frac{23}{2})\)		not available
\(L(1)\)		not available

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