LMFDB - L-function 2-75-5.4-c17-0-3

Dirichlet series

L(s) = 1

+ 442. i·2^-s + 6.56e3i·3^-s − 6.47e4·4^-s − 2.90e6·6^-s + 2.47e7i·7^-s + 2.93e7i·8^-s − 4.30e7·9^-s − 7.66e6·11^-s − 4.25e8i·12^-s + 3.40e9i·13^-s − 1.09e10·14^-s − 2.14e10·16^-s − 5.35e10i·17^-s − 1.90e10i·18^-s + 1.29e11·19^-s + ⋯

L(s) = 1

+ 1.22i·2^-s + 0.577i·3^-s − 0.494·4^-s − 0.705·6^-s + 1.62i·7^-s + 0.618i·8^-s − 0.333·9^-s − 0.0107·11^-s − 0.285i·12^-s + 1.15i·13^-s − 1.98·14^-s − 1.24·16^-s − 1.86i·17^-s − 0.407i·18^-s + 1.75·19^-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

$L(9)$	$\approx$	$0.5559853770$
$L(\frac12)$	$\approx$	$0.5559853770$
$L(\frac{19}{2})$		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 - 442. iT - 1.31e5T^{2} $
	7	$ 1 - 2.47e7iT - 2.32e14T^{2} $
	11	$ 1 + 7.66e6T + 5.05e17T^{2} $
	13	$ 1 - 3.40e9iT - 8.65e18T^{2} $
	17	$ 1 + 5.35e10iT - 8.27e20T^{2} $
	19	$ 1 - 1.29e11T + 5.48e21T^{2} $
	23	$ 1 + 2.11e11iT - 1.41e23T^{2} $
	29	$ 1 + 4.13e12T + 7.25e24T^{2} $
	31	$ 1 + 5.31e12T + 2.25e25T^{2} $
	37	$ 1 - 1.85e13iT - 4.56e26T^{2} $
	41	$ 1 + 5.57e13T + 2.61e27T^{2} $
	43	$ 1 - 7.31e13iT - 5.87e27T^{2} $
	47	$ 1 + 1.22e14iT - 2.66e28T^{2} $
	53	$ 1 + 2.74e14iT - 2.05e29T^{2} $
	59	$ 1 + 6.31e14T + 1.27e30T^{2} $
	61	$ 1 + 1.14e15T + 2.24e30T^{2} $
	67	$ 1 - 6.08e14iT - 1.10e31T^{2} $
	71	$ 1 - 1.25e15T + 2.96e31T^{2} $
	73	$ 1 + 7.71e15iT - 4.74e31T^{2} $
	79	$ 1 - 2.63e16T + 1.81e32T^{2} $
	83	$ 1 + 1.45e16iT - 4.21e32T^{2} $
	89	$ 1 + 2.61e16T + 1.37e33T^{2} $
	97	$ 1 + 9.64e15iT - 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

−11.86652506551090012840280662781, −11.43776670420843438141086250917, −9.408494732447589939939024996256, −9.050104068032412107916511838201, −7.70587037138289900334487433650, −6.62503005704474734433785193377, −5.44454859742561201110356998228, −4.93910996920203595078030840741, −3.10105376187411707214717213892, −1.98917991458687458124530985468, 0.11085298652654882471058409741, 1.07450187290395999169108441349, 1.75024239825856616380356573419, 3.31709581207161144211834516894, 3.85981547870517862420178711263, 5.61301962593458087802392794341, 7.10569732005051181091005180512, 7.83189205493741231292207179489, 9.527310321217954411945693651668, 10.56441888439374447936372508006

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

\(L(9)\)	\(\approx\)	\(0.5559853770\)
\(L(\frac12)\)	\(\approx\)	\(0.5559853770\)
\(L(\frac{19}{2})\)		not available
\(L(1)\)		not available

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