LMFDB - L-function 2-4100-5.4-c1-0-1

Dirichlet series

L(s) = 1

− 3.41i·3^-s − 2.44i·7^-s − 8.68·9^-s − 3.18·11^-s − 2.56i·13^-s + 3.03i·17^-s − 5.23·19^-s − 8.35·21^-s + 6.06i·23^-s + 19.4i·27^-s + 3.49·29^-s + 2.10·31^-s + 10.8i·33^-s + 11.7i·37^-s − 8.77·39^-s + ⋯

L(s) = 1

− 1.97i·3^-s − 0.923i·7^-s − 2.89·9^-s − 0.959·11^-s − 0.711i·13^-s + 0.735i·17^-s − 1.20·19^-s − 1.82·21^-s + 1.26i·23^-s + 3.73i·27^-s + 0.648·29^-s + 0.377·31^-s + 1.89i·33^-s + 1.92i·37^-s − 1.40·39^-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

$L(1)$	$\approx$	$0.1941522387$
$L(\frac12)$	$\approx$	$0.1941522387$
$L(\frac{3}{2})$		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 + 3.41iT - 3T^{2} $
	7	$ 1 + 2.44iT - 7T^{2} $
	11	$ 1 + 3.18T + 11T^{2} $
	13	$ 1 + 2.56iT - 13T^{2} $
	17	$ 1 - 3.03iT - 17T^{2} $
	19	$ 1 + 5.23T + 19T^{2} $
	23	$ 1 - 6.06iT - 23T^{2} $
	29	$ 1 - 3.49T + 29T^{2} $
	31	$ 1 - 2.10T + 31T^{2} $
	37	$ 1 - 11.7iT - 37T^{2} $
	43	$ 1 + 11.0iT - 43T^{2} $
	47	$ 1 + 3.36iT - 47T^{2} $
	53	$ 1 - 1.72iT - 53T^{2} $
	59	$ 1 + 5.43T + 59T^{2} $
	61	$ 1 - 9.91T + 61T^{2} $
	67	$ 1 + 11.4iT - 67T^{2} $
	71	$ 1 + 0.455T + 71T^{2} $
	73	$ 1 + 11.9iT - 73T^{2} $
	79	$ 1 + 4.85T + 79T^{2} $
	83	$ 1 - 4.15iT - 83T^{2} $
	89	$ 1 + 12.0T + 89T^{2} $
	97	$ 1 + 0.605iT - 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

−8.195374325965896478714394169585, −7.74515594004300431497096948450, −7.07613776846837931377548449040, −6.45158493266668732914891563564, −5.76625794587476053163748449947, −4.96501219369342919928373597458, −3.64147646899747760220686351784, −2.77578189065325731584688775408, −1.89364429567676710329282998071, −0.982452559464454804832915215778, 0.06118248925299868575978179022, 2.56787585291729724390694133182, 2.68984362682801181294391290851, 4.08077873284936402734994255872, 4.48526198289851700032720962894, 5.28512208141468225933443869323, 5.84939298210935234058090394239, 6.69806211676139312857819812753, 8.040223564761769784309020419852, 8.623429587301826485260047495506

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1941522387\)
\(L(\frac12)\)	\(\approx\)	\(0.1941522387\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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