LMFDB - L-function 2-4140-345.344-c1-0-22

Dirichlet series

L(s) = 1

+ 2.23i·5^-s − 5.21·7^-s + 3.41i·17^-s − 4.79i·23^-s − 5.00·25^-s + 1.80i·29^-s + 2.76·31^-s − 11.6i·35^-s − 9.55·37^-s − 8.54i·41^-s − 13.1·43^-s + 20.1·49^-s + 13.5i·53^-s + 13.4i·59^-s + 16.3·67^-s + ⋯

L(s) = 1

+ 0.999i·5^-s − 1.96·7^-s + 0.827i·17^-s − 0.999i·23^-s − 1.00·25^-s + 0.336i·29^-s + 0.496·31^-s − 1.96i·35^-s − 1.57·37^-s − 1.33i·41^-s − 1.99·43^-s + 2.87·49^-s + 1.86i·53^-s + 1.74i·59^-s + 1.99·67^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4140$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$33.0580$
Root analytic conductor:	$5.74961$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4140} (2069, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4140,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$0.6476164334$
$L(\frac12)$	$\approx$	$0.6476164334$
$L(\frac{3}{2})$		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 - 2.23iT $
	23	$ 1 + 4.79iT $
good	7	$ 1 + 5.21T + 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 3.41iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	29	$ 1 - 1.80iT - 29T^{2} $
	31	$ 1 - 2.76T + 31T^{2} $
	37	$ 1 + 9.55T + 37T^{2} $
	41	$ 1 + 8.54iT - 41T^{2} $
	43	$ 1 + 13.1T + 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 13.5iT - 53T^{2} $
	59	$ 1 - 13.4iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 16.3T + 67T^{2} $
	71	$ 1 + 9.78iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 + 18.0iT - 83T^{2} $
	89	$ 1 + 89T^{2} $
	97	$ 1 - 16.7T + 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

−8.412910066613809117534663295293, −7.26239289557642402739465193576, −6.83207619851118599288701552500, −6.22651856928903577844326629498, −5.63941729977272207178658287525, −4.32094284967228159213843711313, −3.43458267438773728405923444570, −3.03485324735752423486116027373, −2.00831192515048039297610489982, −0.24696108383167413152660432668, 0.78056364712549526577007311188, 2.13335656380609133224362272104, 3.30666591682753094022503377091, 3.72289014302932144522218575898, 4.96274322826989681306700896305, 5.44938106851919631533624988190, 6.53992122961901129607318090226, 6.79262863451105588158561246550, 7.910660985109989237932994495656, 8.570085256846347891331095545881

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6476164334\)
\(L(\frac12)\)	\(\approx\)	\(0.6476164334\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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