LMFDB - L-function 2-2960-740.739-c0-0-0

Dirichlet series

L(s) = 1

− 0.765·3^-s − 5^-s − 1.84·7^-s − 0.414·9^-s − 1.41·13^-s + 0.765·15^-s + 1.41·17^-s − 0.765·19^-s + 1.41·21^-s + 25^-s + 1.08·27^-s − 1.84·31^-s + 1.84·35^-s + 37^-s + 1.08·39^-s + 0.414·45^-s + 1.84·47^-s + 2.41·49^-s − 1.08·51^-s + 0.585·57^-s − 1.84·59^-s + 0.765·63^-s + 1.41·65^-s + 0.765·67^-s − 0.765·75^-s + 1.84·79^-s − 0.414·81^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2960$ = $2^{4} \cdot 5 \cdot 37$
Sign:	$1$
Analytic conductor:	$1.47723$
Root analytic conductor:	$1.21541$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2960} (2959, \cdot )$
Primitive:	yes
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(2,\ 2960,\ (\ :0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3344958353$
$L(\frac12)$	$\approx$	$0.3344958353$
$L(1)$		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 + T $
	37	$ 1 - T $
good	3	$ 1 + 0.765T + T^{2} $
	7	$ 1 + 1.84T + T^{2} $
	11	$ 1 - T^{2} $
	13	$ 1 + 1.41T + T^{2} $
	17	$ 1 - 1.41T + T^{2} $
	19	$ 1 + 0.765T + T^{2} $
	23	$ 1 - T^{2} $
	29	$ 1 - T^{2} $
	31	$ 1 + 1.84T + T^{2} $
	41	$ 1 + T^{2} $
	43	$ 1 - T^{2} $
	47	$ 1 - 1.84T + T^{2} $
	53	$ 1 - T^{2} $
	59	$ 1 + 1.84T + T^{2} $
	61	$ 1 - T^{2} $
	67	$ 1 - 0.765T + T^{2} $
	71	$ 1 - T^{2} $
	73	$ 1 - T^{2} $
	79	$ 1 - 1.84T + T^{2} $
	83	$ 1 - 0.765T + T^{2} $
	89	$ 1 - T^{2} $
	97	$ 1 + T^{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

−9.151901594014824773834083681263, −8.000838280462582383550883350191, −7.32502923261084964438607815942, −6.67909327655731834471803967769, −5.87615811484858734475111059774, −5.22552537334318415479525002581, −4.14193346383975944259031030981, −3.33737701116877980251195144040, −2.60076582922549783578647103272, −0.50549037690951651395956898947, 0.50549037690951651395956898947, 2.60076582922549783578647103272, 3.33737701116877980251195144040, 4.14193346383975944259031030981, 5.22552537334318415479525002581, 5.87615811484858734475111059774, 6.67909327655731834471803967769, 7.32502923261084964438607815942, 8.000838280462582383550883350191, 9.151901594014824773834083681263

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.3344958353\)
\(L(\frac12)\)	\(\approx\)	\(0.3344958353\)
\(L(1)\)		not available
\(L(1)\)		not available

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