LMFDB - L-function 2-8280-69.68-c1-0-12

Dirichlet series

L(s) = 1

− 5^-s + 1.44i·7^-s − 1.36·11^-s − 1.07·13^-s − 0.324·17^-s + 0.711i·19^-s + (−0.999 − 4.69i)23^-s + 25^-s + 2.07i·29^-s + 5.82·31^-s − 1.44i·35^-s − 6.84i·37^-s + 1.86i·41^-s + 5.81i·43^-s − 12.1i·47^-s + ⋯

L(s) = 1

− 0.447·5^-s + 0.546i·7^-s − 0.410·11^-s − 0.298·13^-s − 0.0787·17^-s + 0.163i·19^-s + (−0.208 − 0.978i)23^-s + 0.200·25^-s + 0.385i·29^-s + 1.04·31^-s − 0.244i·35^-s − 1.12i·37^-s + 0.291i·41^-s + 0.887i·43^-s − 1.77i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8280$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$-0.734 - 0.678i$
Analytic conductor:	$66.1161$
Root analytic conductor:	$8.13118$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8280} (1241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8280,\ (\ :1/2),\ -0.734 - 0.678i)$

Particular Values

$L(1)$	$\approx$	$0.7009534136$
$L(\frac12)$	$\approx$	$0.7009534136$
$L(\frac{3}{2})$		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 $
	5	$ 1 + T $
	23	$ 1 + (0.999 + 4.69i)T $
good	7	$ 1 - 1.44iT - 7T^{2} $
	11	$ 1 + 1.36T + 11T^{2} $
	13	$ 1 + 1.07T + 13T^{2} $
	17	$ 1 + 0.324T + 17T^{2} $
	19	$ 1 - 0.711iT - 19T^{2} $
	29	$ 1 - 2.07iT - 29T^{2} $
	31	$ 1 - 5.82T + 31T^{2} $
	37	$ 1 + 6.84iT - 37T^{2} $
	41	$ 1 - 1.86iT - 41T^{2} $
	43	$ 1 - 5.81iT - 43T^{2} $
	47	$ 1 + 12.1iT - 47T^{2} $
	53	$ 1 - 7.33T + 53T^{2} $
	59	$ 1 - 9.74iT - 59T^{2} $
	61	$ 1 - 9.45iT - 61T^{2} $
	67	$ 1 - 9.90iT - 67T^{2} $
	71	$ 1 - 13.5iT - 71T^{2} $
	73	$ 1 + 4.12T + 73T^{2} $
	79	$ 1 + 11.4iT - 79T^{2} $
	83	$ 1 - 0.800T + 83T^{2} $
	89	$ 1 + 8.45T + 89T^{2} $
	97	$ 1 - 8.16iT - 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

−8.190412126891768674426645102395, −7.31004079056284749861661440888, −6.83505647773143220448330233034, −5.88178367104836020417910971312, −5.38024008857241655184097961074, −4.47890512410066951261852674022, −3.91683530805089434064500675523, −2.80212910098265346420976093075, −2.34649660295967480829790560722, −1.03533521845135876971233883743, 0.18641130407975634116144424205, 1.28623382484548044699221910513, 2.39951957550830454796529946734, 3.23699481733223949405824400486, 3.99214068973527527803892370966, 4.70947695037453020981068110169, 5.36135405168228880091005022168, 6.27608736880448242343901682726, 6.89185122403320076744387764640, 7.69112675738236810236588609299

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7009534136\)
\(L(\frac12)\)	\(\approx\)	\(0.7009534136\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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