LMFDB - L-function 2-4100-5.4-c1-0-14

Dirichlet series

L(s) = 1

− 2.47i·3^-s + 0.154i·7^-s − 3.14·9^-s − 4.66·11^-s + 6.51i·13^-s − 0.170i·17^-s − 1.98·19^-s + 0.382·21^-s − 0.648i·23^-s + 0.347i·27^-s + 2.94·29^-s − 3.23·31^-s + 11.5i·33^-s + 0.224i·37^-s + 16.1·39^-s + ⋯

L(s) = 1

− 1.43i·3^-s + 0.0582i·7^-s − 1.04·9^-s − 1.40·11^-s + 1.80i·13^-s − 0.0412i·17^-s − 0.455·19^-s + 0.0833·21^-s − 0.135i·23^-s + 0.0668i·27^-s + 0.546·29^-s − 0.580·31^-s + 2.01i·33^-s + 0.0369i·37^-s + 2.58·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4100$ = $2^{2} \cdot 5^{2} \cdot 41$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$32.7386$
Root analytic conductor:	$5.72177$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4100} (1149, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4100,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.410342059$
$L(\frac12)$	$\approx$	$1.410342059$
$L(\frac{3}{2})$		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 $
	41	$ 1 - T $
good	3	$ 1 + 2.47iT - 3T^{2} $
	7	$ 1 - 0.154iT - 7T^{2} $
	11	$ 1 + 4.66T + 11T^{2} $
	13	$ 1 - 6.51iT - 13T^{2} $
	17	$ 1 + 0.170iT - 17T^{2} $
	19	$ 1 + 1.98T + 19T^{2} $
	23	$ 1 + 0.648iT - 23T^{2} $
	29	$ 1 - 2.94T + 29T^{2} $
	31	$ 1 + 3.23T + 31T^{2} $
	37	$ 1 - 0.224iT - 37T^{2} $
	43	$ 1 + 1.55iT - 43T^{2} $
	47	$ 1 + 0.959iT - 47T^{2} $
	53	$ 1 + 3.23iT - 53T^{2} $
	59	$ 1 - 12.5T + 59T^{2} $
	61	$ 1 - 13.0T + 61T^{2} $
	67	$ 1 + 2.84iT - 67T^{2} $
	71	$ 1 - 2.61T + 71T^{2} $
	73	$ 1 - 13.5iT - 73T^{2} $
	79	$ 1 - 8.14T + 79T^{2} $
	83	$ 1 + 2.02iT - 83T^{2} $
	89	$ 1 - 10.7T + 89T^{2} $
	97	$ 1 + 7.77iT - 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

−8.329559643603607771114150165659, −7.48280875180240851257742883709, −6.97902886792577458284499638541, −6.39610244000834481107606796864, −5.54811665451238962389675958985, −4.70433090640901084290213282811, −3.71469682125180651476668307592, −2.34365055480449810934824930445, −2.10811434152339299241795915541, −0.796378900889067020582834116605, 0.54160768879524495140474930509, 2.38989935125814429406876079209, 3.15488078945557633459240756647, 3.87617922223995290320566169034, 4.83685498157616395659826200589, 5.34501412017831096477162519351, 5.91387020732056683702152954814, 7.15137874575934964388547343989, 7.963390192898614227155819907366, 8.451855475463679649580621298123

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.410342059\)
\(L(\frac12)\)	\(\approx\)	\(1.410342059\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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