LMFDB - L-function 2-1152-8.5-c1-0-18

Dirichlet series

L(s) = 1

− 4i·5^-s − 4i·13^-s + 2·17^-s − 11·25^-s + 4i·29^-s − 12i·37^-s − 10·41^-s − 7·49^-s − 4i·53^-s + 12i·61^-s − 16·65^-s + 6·73^-s − 8i·85^-s + 10·89^-s − 18·97^-s + ⋯

L(s) = 1

− 1.78i·5^-s − 1.10i·13^-s + 0.485·17^-s − 2.20·25^-s + 0.742i·29^-s − 1.97i·37^-s − 1.56·41^-s − 49^-s − 0.549i·53^-s + 1.53i·61^-s − 1.98·65^-s + 0.702·73^-s − 0.867i·85^-s + 1.05·89^-s − 1.82·97^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$9.19876$
Root analytic conductor:	$3.03294$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.272987367$
$L(\frac12)$	$\approx$	$1.272987367$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
good	5	$ 1 + 4iT - 5T^{2} $
	7	$ 1 + 7T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 + 4iT - 13T^{2} $
	17	$ 1 - 2T + 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 4iT - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 12iT - 37T^{2} $
	41	$ 1 + 10T + 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 + 4iT - 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 - 12iT - 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 - 10T + 89T^{2} $
	97	$ 1 + 18T + 97T^{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

−9.412983801914120450635139908670, −8.596372163455626747867835603023, −8.097085680741996873427802886067, −7.14743271254536446673118724827, −5.74708433530533779237674016995, −5.28879479914129089592170942056, −4.38572120862416764966125962303, −3.30944767592254057123366919533, −1.72891638759806007269784398349, −0.55054134512337012215872817462, 1.87866037799222118240539755892, 2.96490978269294128266084751993, 3.75548881526710708205107976093, 4.94972216285662246032357953441, 6.30415222487800883221735704040, 6.62494658530065983166575515696, 7.52229618763074321093200359068, 8.331103270692023425713997314543, 9.574278187733444783765859485145, 10.06929711786938887413013736139

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.272987367\)
\(L(\frac12)\)	\(\approx\)	\(1.272987367\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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