LMFDB - L-function 2-1216-152.75-c1-0-34

Dirichlet series

L(s) = 1

− 4.35i·5^-s − 3i·7^-s + 3·9^-s − 4.35·11^-s + 7·17^-s + 4.35·19^-s − 4i·23^-s − 14.0·25^-s − 13.0·35^-s − 13.0·43^-s − 13.0i·45^-s + 13i·47^-s − 2·49^-s + 19.0i·55^-s − 4.35i·61^-s + ⋯

L(s) = 1

− 1.94i·5^-s − 1.13i·7^-s + 9^-s − 1.31·11^-s + 1.69·17^-s + 1.00·19^-s − 0.834i·23^-s − 2.80·25^-s − 2.21·35^-s − 1.99·43^-s − 1.94i·45^-s + 1.89i·47^-s − 0.285·49^-s + 2.56i·55^-s − 0.558i·61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1216$ = $2^{6} \cdot 19$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$9.70980$
Root analytic conductor:	$3.11605$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1216} (607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1216,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.546655545$
$L(\frac12)$	$\approx$	$1.546655545$
$L(\frac{3}{2})$		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 $
	19	$ 1 - 4.35T $
good	3	$ 1 - 3T^{2} $
	5	$ 1 + 4.35iT - 5T^{2} $
	7	$ 1 + 3iT - 7T^{2} $
	11	$ 1 + 4.35T + 11T^{2} $
	13	$ 1 + 13T^{2} $
	17	$ 1 - 7T + 17T^{2} $
	23	$ 1 + 4iT - 23T^{2} $
	29	$ 1 + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 13.0T + 43T^{2} $
	47	$ 1 - 13iT - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 + 4.35iT - 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 11T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 - 8.71T + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 - 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

−9.749393363677848593871676207406, −8.446862728355132812772884625237, −7.82853368535586807523609106040, −7.28019754241498009659040552412, −5.81263620916502455403853978296, −4.96565149590601462395531688317, −4.47050555008448355340762828076, −3.36368190604318474366024854417, −1.52586310849451429400949758104, −0.69911046023946851255477402298, 1.93680931932205370632783871178, 2.97723176560102034372782779674, 3.52967760534338542679190086557, 5.23385125515505769292448144974, 5.78458890582269204569409955706, 6.88805845467818410222614558906, 7.49634751387722474961562226909, 8.133096166120605187853632754073, 9.570762131679372410567805352438, 10.11585355743316227076640962654

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.546655545\)
\(L(\frac12)\)	\(\approx\)	\(1.546655545\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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