LMFDB - L-function 2-3380-13.12-c1-0-24

Dirichlet series

L(s) = 1

− 0.0947·3^-s + i·5^-s + 0.826i·7^-s − 2.99·9^-s − 1.73i·11^-s − 0.0947i·15^-s − 1.43·17^-s − 1.06i·19^-s − 0.0783i·21^-s − 3.08·23^-s − 25^-s + 0.567·27^-s + 7.45·29^-s − 5.84i·31^-s + 0.164i·33^-s + ⋯

L(s) = 1

− 0.0547·3^-s + 0.447i·5^-s + 0.312i·7^-s − 0.997·9^-s − 0.522i·11^-s − 0.0244i·15^-s − 0.347·17^-s − 0.245i·19^-s − 0.0171i·21^-s − 0.643·23^-s − 0.200·25^-s + 0.109·27^-s + 1.38·29^-s − 1.04i·31^-s + 0.0285i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3380$ = $2^{2} \cdot 5 \cdot 13^{2}$
Sign:	$0.960 + 0.277i$
Analytic conductor:	$26.9894$
Root analytic conductor:	$5.19513$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3380} (3041, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3380,\ (\ :1/2),\ 0.960 + 0.277i)$

Particular Values

$L(1)$	$\approx$	$1.428298475$
$L(\frac12)$	$\approx$	$1.428298475$
$L(\frac{3}{2})$		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 $
	5	$ 1 - iT $
	13	$ 1 $
good	3	$ 1 + 0.0947T + 3T^{2} $
	7	$ 1 - 0.826iT - 7T^{2} $
	11	$ 1 + 1.73iT - 11T^{2} $
	17	$ 1 + 1.43T + 17T^{2} $
	19	$ 1 + 1.06iT - 19T^{2} $
	23	$ 1 + 3.08T + 23T^{2} $
	29	$ 1 - 7.45T + 29T^{2} $
	31	$ 1 + 5.84iT - 31T^{2} $
	37	$ 1 + 0.983iT - 37T^{2} $
	41	$ 1 - 4.26iT - 41T^{2} $
	43	$ 1 - 9.54T + 43T^{2} $
	47	$ 1 + 3.46iT - 47T^{2} $
	53	$ 1 - 0.334T + 53T^{2} $
	59	$ 1 + 11.5iT - 59T^{2} $
	61	$ 1 - 2.71T + 61T^{2} $
	67	$ 1 - 13.7iT - 67T^{2} $
	71	$ 1 - 9.77iT - 71T^{2} $
	73	$ 1 + 11.1iT - 73T^{2} $
	79	$ 1 + 0.252T + 79T^{2} $
	83	$ 1 + 5.67iT - 83T^{2} $
	89	$ 1 + 4.59iT - 89T^{2} $
	97	$ 1 - 9.53iT - 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

−8.549297414373189541712116451881, −7.960194457322392265587749724545, −7.05403505046409445337089134079, −6.15603271755888049251567965658, −5.78757913399381729395698678397, −4.78008991258862861431691273306, −3.82235474113957327846158516488, −2.87100313799707752753737409429, −2.23353499342802024713282035244, −0.59516300462321637334736266986, 0.816829425341699260314534919163, 2.09906875004037978981109085270, 3.03584033439023002778487726221, 4.08513420701201320816498788324, 4.79463987124786041052747031498, 5.64019697300556025237972940995, 6.33733453114971975882530829556, 7.20222207658697134349230030594, 7.972332597723095953986475589717, 8.684171471491328488395489766266

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.428298475\)
\(L(\frac12)\)	\(\approx\)	\(1.428298475\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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