LMFDB - L-function 2-75-5.4-c17-0-6

Dirichlet series

L(s) = 1

− 670. i·2^-s + 6.56e3i·3^-s − 3.18e5·4^-s + 4.39e6·6^-s − 2.70e7i·7^-s + 1.25e8i·8^-s − 4.30e7·9^-s + 1.01e9·11^-s − 2.09e9i·12^-s + 1.04e9i·13^-s − 1.81e10·14^-s + 4.25e10·16^-s + 2.00e10i·17^-s + 2.88e10i·18^-s − 1.64e10·19^-s + ⋯

L(s) = 1

− 1.85i·2^-s + 0.577i·3^-s − 2.43·4^-s + 1.06·6^-s − 1.77i·7^-s + 2.65i·8^-s − 0.333·9^-s + 1.42·11^-s − 1.40i·12^-s + 0.356i·13^-s − 3.28·14^-s + 2.47·16^-s + 0.695i·17^-s + 0.617i·18^-s − 0.222·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$75$ = $3 \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$137.416$
Root analytic conductor:	$11.7224$
Motivic weight:	$17$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{75} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 75,\ (\ :17/2),\ 0.894 + 0.447i)$

Particular Values

$L(9)$	$\approx$	$0.8464354716$
$L(\frac12)$	$\approx$	$0.8464354716$
$L(\frac{19}{2})$		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 - 6.56e3iT $
	5	$ 1 $
good	2	$ 1 + 670. iT - 1.31e5T^{2} $
	7	$ 1 + 2.70e7iT - 2.32e14T^{2} $
	11	$ 1 - 1.01e9T + 5.05e17T^{2} $
	13	$ 1 - 1.04e9iT - 8.65e18T^{2} $
	17	$ 1 - 2.00e10iT - 8.27e20T^{2} $
	19	$ 1 + 1.64e10T + 5.48e21T^{2} $
	23	$ 1 + 5.92e11iT - 1.41e23T^{2} $
	29	$ 1 + 4.83e11T + 7.25e24T^{2} $
	31	$ 1 + 5.20e12T + 2.25e25T^{2} $
	37	$ 1 - 3.82e13iT - 4.56e26T^{2} $
	41	$ 1 + 6.56e13T + 2.61e27T^{2} $
	43	$ 1 - 5.83e13iT - 5.87e27T^{2} $
	47	$ 1 - 7.43e12iT - 2.66e28T^{2} $
	53	$ 1 - 3.01e14iT - 2.05e29T^{2} $
	59	$ 1 - 3.44e14T + 1.27e30T^{2} $
	61	$ 1 + 2.43e15T + 2.24e30T^{2} $
	67	$ 1 - 1.19e15iT - 1.10e31T^{2} $
	71	$ 1 - 5.42e15T + 2.96e31T^{2} $
	73	$ 1 - 5.84e15iT - 4.74e31T^{2} $
	79	$ 1 + 2.03e15T + 1.81e32T^{2} $
	83	$ 1 - 1.18e16iT - 4.21e32T^{2} $
	89	$ 1 - 2.43e16T + 1.37e33T^{2} $
	97	$ 1 + 1.06e17iT - 5.95e33T^{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

−10.95184350496223902099335303278, −10.33421120434751738659039399931, −9.463824690202077841972183817260, −8.387974134280039249100518106084, −6.63180365032451352367206597920, −4.59115642816428693679541230907, −4.06478888634505481129645612234, −3.26010324502087607586258006061, −1.68736402943775546471103914485, −0.902730941519237865559893619735, 0.20868788133135048411350741991, 1.80706405796626756300877193467, 3.56853914523764386412216337877, 5.20927350530011611641417250358, 5.83618451595910146860688654436, 6.78430647952541454918627684399, 7.78086712602719985354918549276, 8.986951519832751400439525155282, 9.269107798520050556372146689987, 11.65370032847245446869005377095

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 $\F_{q}$
Belyi maps

\(L(9)\)	\(\approx\)	\(0.8464354716\)
\(L(\frac12)\)	\(\approx\)	\(0.8464354716\)
\(L(\frac{19}{2})\)		not available
\(L(1)\)		not available

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