LMFDB - L-function 2-90e2-5.4-c1-0-64

Dirichlet series

L(s) = 1

+ 0.680i·7^-s − 1.68·11^-s − 5.14i·13^-s − 1.31i·17^-s + 0.324·19^-s − 3.78i·23^-s + 8.64·29^-s − 4.14·31^-s + 1.35i·37^-s − 7.15·41^-s + 7.29i·43^-s − 12.9i·47^-s + 6.53·49^-s + 8.83i·53^-s − 8.81·59^-s + ⋯

L(s) = 1

+ 0.257i·7^-s − 0.506·11^-s − 1.42i·13^-s − 0.320i·17^-s + 0.0744·19^-s − 0.790i·23^-s + 1.60·29^-s − 0.744·31^-s + 0.222i·37^-s − 1.11·41^-s + 1.11i·43^-s − 1.89i·47^-s + 0.933·49^-s + 1.21i·53^-s − 1.14·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8100$ = $2^{2} \cdot 3^{4} \cdot 5^{2}$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$64.6788$
Root analytic conductor:	$8.04231$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8100} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8100,\ (\ :1/2),\ -0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$0.7017396928$
$L(\frac12)$	$\approx$	$0.7017396928$
$L(\frac{3}{2})$		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 $
good	7	$ 1 - 0.680iT - 7T^{2} $
	11	$ 1 + 1.68T + 11T^{2} $
	13	$ 1 + 5.14iT - 13T^{2} $
	17	$ 1 + 1.31iT - 17T^{2} $
	19	$ 1 - 0.324T + 19T^{2} $
	23	$ 1 + 3.78iT - 23T^{2} $
	29	$ 1 - 8.64T + 29T^{2} $
	31	$ 1 + 4.14T + 31T^{2} $
	37	$ 1 - 1.35iT - 37T^{2} $
	41	$ 1 + 7.15T + 41T^{2} $
	43	$ 1 - 7.29iT - 43T^{2} $
	47	$ 1 + 12.9iT - 47T^{2} $
	53	$ 1 - 8.83iT - 53T^{2} $
	59	$ 1 + 8.81T + 59T^{2} $
	61	$ 1 - 9.97T + 61T^{2} $
	67	$ 1 + 4.17iT - 67T^{2} $
	71	$ 1 + 0.891T + 71T^{2} $
	73	$ 1 - 7.82iT - 73T^{2} $
	79	$ 1 + 9.65T + 79T^{2} $
	83	$ 1 + 4.85iT - 83T^{2} $
	89	$ 1 + 17.4T + 89T^{2} $
	97	$ 1 - 8.93iT - 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

−7.55633418833567952825090093388, −6.87738234676263169385016873771, −6.10418354338057897387887740700, −5.36916428733974695637243988289, −4.89014462755176294491250837122, −3.93316166853753179358049869080, −2.96098619435775293763046943899, −2.54277107848030204579323441628, −1.24333031084701831105976292326, −0.16916448957599539670501849642, 1.26740340157108074234737462287, 2.10364954144509033376706359870, 3.04742374606549967369308139248, 3.92423699424195632169962309701, 4.54182578306732549940314550730, 5.32912134391921481839775764494, 6.06445311920762762703548016599, 6.90142411554632798815829019087, 7.24732800061102765245454887802, 8.207926347362846662058168504831

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.7017396928\)
\(L(\frac12)\)	\(\approx\)	\(0.7017396928\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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