LMFDB - L-function 2-1040-13.12-c1-0-24

Dirichlet series

L(s) = 1

+ 2.80·3^-s − i·5^-s − 3.54i·7^-s + 4.88·9^-s + 1.26i·11^-s + (2.80 − 2.26i)13^-s − 2.80i·15^-s − 5.42·17^-s + 0.926i·19^-s − 9.95i·21^-s + 7.33·23^-s − 25^-s + 5.27·27^-s − 7.49·29^-s + 8.16i·31^-s + ⋯

L(s) = 1

+ 1.62·3^-s − 0.447i·5^-s − 1.33i·7^-s + 1.62·9^-s + 0.380i·11^-s + (0.778 − 0.627i)13^-s − 0.724i·15^-s − 1.31·17^-s + 0.212i·19^-s − 2.17i·21^-s + 1.52·23^-s − 0.200·25^-s + 1.01·27^-s − 1.39·29^-s + 1.46i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1040$ = $2^{4} \cdot 5 \cdot 13$
Sign:	$0.627 + 0.778i$
Analytic conductor:	$8.30444$
Root analytic conductor:	$2.88174$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1040} (961, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1040,\ (\ :1/2),\ 0.627 + 0.778i)$

Particular Values

$L(1)$	$\approx$	$2.802591928$
$L(\frac12)$	$\approx$	$2.802591928$
$L(\frac{3}{2})$		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 + (-2.80 + 2.26i)T $
good	3	$ 1 - 2.80T + 3T^{2} $
	7	$ 1 + 3.54iT - 7T^{2} $
	11	$ 1 - 1.26iT - 11T^{2} $
	17	$ 1 + 5.42T + 17T^{2} $
	19	$ 1 - 0.926iT - 19T^{2} $
	23	$ 1 - 7.33T + 23T^{2} $
	29	$ 1 + 7.49T + 29T^{2} $
	31	$ 1 - 8.16iT - 31T^{2} $
	37	$ 1 + 7.15iT - 37T^{2} $
	41	$ 1 + 2.18iT - 41T^{2} $
	43	$ 1 - 7.14T + 43T^{2} $
	47	$ 1 + 7.30iT - 47T^{2} $
	53	$ 1 + 2.18T + 53T^{2} $
	59	$ 1 - 4.68iT - 59T^{2} $
	61	$ 1 - 10.2T + 61T^{2} $
	67	$ 1 - 9.83iT - 67T^{2} $
	71	$ 1 - 14.1iT - 71T^{2} $
	73	$ 1 - 0.980iT - 73T^{2} $
	79	$ 1 - 12.1T + 79T^{2} $
	83	$ 1 + 1.15iT - 83T^{2} $
	89	$ 1 - 11.0iT - 89T^{2} $
	97	$ 1 - 14.1iT - 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

−9.559849185561941611549809576352, −8.893659221089772824177273027695, −8.311302842472908249547689141002, −7.32764247360830296927643280006, −6.91871679468654480981615625084, −5.30882522406345796305094579675, −4.10897602978795886239186093195, −3.64546611894607893076788489147, −2.40973671182103259650605983927, −1.15290145310598340922671319509, 1.91175198875274246273140187940, 2.67068506187715796463872270897, 3.50402891954443408718016181871, 4.58502043105368032923687238400, 5.92519905362934065129036109303, 6.76681017680134880698436232977, 7.78317671734897004918127272859, 8.561404689086880262566909053444, 9.191579529910819730722658007621, 9.487054245060235745881954383042

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.802591928\)
\(L(\frac12)\)	\(\approx\)	\(2.802591928\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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