LMFDB - L-function 2-4100-5.4-c1-0-52

Dirichlet series

L(s) = 1

− 2.32i·3^-s − 2.51i·7^-s − 2.41·9^-s + 3.66·11^-s + 3.06i·13^-s − 7.33i·17^-s + 3.55·19^-s − 5.85·21^-s + 2.14i·23^-s − 1.35i·27^-s − 5.03·29^-s + 3.96·31^-s − 8.52i·33^-s − 5.71i·37^-s + 7.13·39^-s + ⋯

L(s) = 1

− 1.34i·3^-s − 0.950i·7^-s − 0.806·9^-s + 1.10·11^-s + 0.849i·13^-s − 1.78i·17^-s + 0.816·19^-s − 1.27·21^-s + 0.447i·23^-s − 0.260i·27^-s − 0.934·29^-s + 0.711·31^-s − 1.48i·33^-s − 0.939i·37^-s + 1.14·39^-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

$L(1)$	$\approx$	$1.936836678$
$L(\frac12)$	$\approx$	$1.936836678$
$L(\frac{3}{2})$		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 + 2.32iT - 3T^{2} $
	7	$ 1 + 2.51iT - 7T^{2} $
	11	$ 1 - 3.66T + 11T^{2} $
	13	$ 1 - 3.06iT - 13T^{2} $
	17	$ 1 + 7.33iT - 17T^{2} $
	19	$ 1 - 3.55T + 19T^{2} $
	23	$ 1 - 2.14iT - 23T^{2} $
	29	$ 1 + 5.03T + 29T^{2} $
	31	$ 1 - 3.96T + 31T^{2} $
	37	$ 1 + 5.71iT - 37T^{2} $
	43	$ 1 - 1.65iT - 43T^{2} $
	47	$ 1 - 0.225iT - 47T^{2} $
	53	$ 1 - 0.892iT - 53T^{2} $
	59	$ 1 - 13.3T + 59T^{2} $
	61	$ 1 - 1.35T + 61T^{2} $
	67	$ 1 + 12.0iT - 67T^{2} $
	71	$ 1 + 8.75T + 71T^{2} $
	73	$ 1 + 10.1iT - 73T^{2} $
	79	$ 1 + 2.02T + 79T^{2} $
	83	$ 1 + 0.808iT - 83T^{2} $
	89	$ 1 - 6.12T + 89T^{2} $
	97	$ 1 - 9.52iT - 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

−7.68978597702694411753377330684, −7.37866510685825212324354707694, −6.83022148067090375658009249770, −6.21661973729670169272198236671, −5.17812067411924392231619735825, −4.25823564229986870109601485136, −3.43664777509515506776613412837, −2.30494177479471840236692661723, −1.36747114566259506143027511453, −0.60925077287777091950919966339, 1.34814400344697908638751726192, 2.60572311060333468003281761875, 3.59010195975033977297021244560, 4.04409349246720270934882064751, 5.01910443105763810420677734100, 5.68243224423891711768743899127, 6.27510563607645344233184432076, 7.29605916189033935984727064845, 8.497353020597969797704064005458, 8.608453856073710314373650804049

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.936836678\)
\(L(\frac12)\)	\(\approx\)	\(1.936836678\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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