LMFDB - L-function 2-75-5.4-c17-0-47

Dirichlet series

L(s) = 1

− 654. i·2^-s − 6.56e3i·3^-s − 2.97e5·4^-s − 4.29e6·6^-s − 3.59e5i·7^-s + 1.08e8i·8^-s − 4.30e7·9^-s + 1.12e9·11^-s + 1.95e9i·12^-s − 3.56e9i·13^-s − 2.35e8·14^-s + 3.22e10·16^-s − 4.86e10i·17^-s + 2.81e10i·18^-s − 5.83e10·19^-s + ⋯

L(s) = 1

− 1.80i·2^-s − 0.577i·3^-s − 2.26·4^-s − 1.04·6^-s − 0.0235i·7^-s + 2.29i·8^-s − 0.333·9^-s + 1.58·11^-s + 1.30i·12^-s − 1.21i·13^-s − 0.0426·14^-s + 1.87·16^-s − 1.68i·17^-s + 0.602i·18^-s − 0.787·19^-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

$L(9)$	$\approx$	$0.9605029689$
$L(\frac12)$	$\approx$	$0.9605029689$
$L(\frac{19}{2})$		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 + 654. iT - 1.31e5T^{2} $
	7	$ 1 + 3.59e5iT - 2.32e14T^{2} $
	11	$ 1 - 1.12e9T + 5.05e17T^{2} $
	13	$ 1 + 3.56e9iT - 8.65e18T^{2} $
	17	$ 1 + 4.86e10iT - 8.27e20T^{2} $
	19	$ 1 + 5.83e10T + 5.48e21T^{2} $
	23	$ 1 + 1.76e11iT - 1.41e23T^{2} $
	29	$ 1 + 2.78e12T + 7.25e24T^{2} $
	31	$ 1 + 6.55e9T + 2.25e25T^{2} $
	37	$ 1 - 2.71e13iT - 4.56e26T^{2} $
	41	$ 1 - 1.87e12T + 2.61e27T^{2} $
	43	$ 1 + 1.01e14iT - 5.87e27T^{2} $
	47	$ 1 + 2.15e14iT - 2.66e28T^{2} $
	53	$ 1 + 8.28e14iT - 2.05e29T^{2} $
	59	$ 1 + 1.11e15T + 1.27e30T^{2} $
	61	$ 1 - 2.77e14T + 2.24e30T^{2} $
	67	$ 1 - 5.09e15iT - 1.10e31T^{2} $
	71	$ 1 + 7.44e15T + 2.96e31T^{2} $
	73	$ 1 - 3.88e14iT - 4.74e31T^{2} $
	79	$ 1 - 1.71e16T + 1.81e32T^{2} $
	83	$ 1 - 7.42e15iT - 4.21e32T^{2} $
	89	$ 1 + 1.06e16T + 1.37e33T^{2} $
	97	$ 1 - 4.95e15iT - 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

−10.40358781034530364615165176014, −9.364561415769279961908467271808, −8.465370474419095727248243497457, −6.90603134485465315654968628074, −5.30661841036734982554791844101, −4.01733014869126090403556724729, −3.01399717736605727211369672314, −1.99763963895179327511617794890, −0.943184619568333872547416813576, −0.23130375151926212844482389084, 1.53646464106336419567958971366, 3.92053799401836260807342683688, 4.36384921027781733302505943621, 5.91386594167575282879959405772, 6.47191168013211733513264064335, 7.69467885992792533326836542056, 8.951955400576414695316938792683, 9.337893431334110962419778504354, 10.96588121019997396547977411992, 12.43829999852074093645606117811

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

\(L(9)\)	\(\approx\)	\(0.9605029689\)
\(L(\frac12)\)	\(\approx\)	\(0.9605029689\)
\(L(\frac{19}{2})\)		not available
\(L(1)\)		not available

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