LMFDB - L-function 2-75-15.14-c8-0-44

Dirichlet series

L(s) = 1

− 81i·3^-s − 256·4^-s − 4.27e3i·7^-s − 6.56e3·9^-s + 2.07e4i·12^-s − 5.64e4i·13^-s + 6.55e4·16^-s − 1.57e5·19^-s − 3.46e5·21^-s + 5.31e5i·27^-s + 1.09e6i·28^-s + 1.22e6·31^-s + 1.67e6·36^-s + 5.03e5i·37^-s − 4.57e6·39^-s + ⋯

L(s) = 1

− i·3^-s − 4^-s − 1.77i·7^-s − 9^-s + i·12^-s − 1.97i·13^-s + 16^-s − 1.21·19^-s − 1.77·21^-s + i·27^-s + 1.77i·28^-s + 1.32·31^-s + 36^-s + 0.268i·37^-s − 1.97·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(9-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$75$ = $3 \cdot 5^{2}$
Sign:	$-0.447 - 0.894i$
Analytic conductor:	$30.5533$
Root analytic conductor:	$5.52751$
Motivic weight:	$8$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{75} (74, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 75,\ (\ :4),\ -0.447 - 0.894i)$

Particular Values

$L(\frac{9}{2})$	$\approx$	$0.350501 + 0.567122i$
$L(\frac12)$	$\approx$	$0.350501 + 0.567122i$
$L(5)$		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 + 81iT $
	5	$ 1 $
good	2	$ 1 + 256T^{2} $
	7	$ 1 + 4.27e3iT - 5.76e6T^{2} $
	11	$ 1 - 2.14e8T^{2} $
	13	$ 1 + 5.64e4iT - 8.15e8T^{2} $
	17	$ 1 + 6.97e9T^{2} $
	19	$ 1 + 1.57e5T + 1.69e10T^{2} $
	23	$ 1 + 7.83e10T^{2} $
	29	$ 1 - 5.00e11T^{2} $
	31	$ 1 - 1.22e6T + 8.52e11T^{2} $
	37	$ 1 - 5.03e5iT - 3.51e12T^{2} $
	41	$ 1 - 7.98e12T^{2} $
	43	$ 1 - 6.83e6iT - 1.16e13T^{2} $
	47	$ 1 + 2.38e13T^{2} $
	53	$ 1 + 6.22e13T^{2} $
	59	$ 1 - 1.46e14T^{2} $
	61	$ 1 + 3.07e5T + 1.91e14T^{2} $
	67	$ 1 + 3.18e7iT - 4.06e14T^{2} $
	71	$ 1 - 6.45e14T^{2} $
	73	$ 1 + 1.61e7iT - 8.06e14T^{2} $
	79	$ 1 - 1.88e7T + 1.51e15T^{2} $
	83	$ 1 + 2.25e15T^{2} $
	89	$ 1 - 3.93e15T^{2} $
	97	$ 1 + 8.21e7iT - 7.83e15T^{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

−12.61984118786648214305499485028, −10.88055418122252282360903537850, −10.00925678448787898912855983818, −8.297493832466086433694683231151, −7.66262366289272289562502119769, −6.23324725414672506786935254052, −4.68175001868724566238201606868, −3.24369769566829045571736169136, −1.03788330250430650622367001582, −0.26006100188008550248040367308, 2.30812251583507723956150837134, 4.01450302712464015884911069848, 5.02792471881898110227390638107, 6.19962336731141950110880663982, 8.638733650037724747393695154884, 8.940340146754462305710987245906, 9.993825920474706422911642402490, 11.50600657062523265195756699583, 12.36066082129345523033311586133, 13.84602224368665758105737761724

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 $\F_{q}$

\(L(\frac{9}{2})\)	\(\approx\)	\(0.350501 + 0.567122i\)
\(L(\frac12)\)	\(\approx\)	\(0.350501 + 0.567122i\)
\(L(5)\)		not available
\(L(1)\)		not available

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