LMFDB - L-function 2-3150-105.104-c1-0-41

Dirichlet series

L(s) = 1

+ 2^-s + 4^-s + (−2.64 − 0.0951i)7^-s + 8^-s + 5.28i·11^-s − 2.19·13^-s + (−2.64 − 0.0951i)14^-s + 16^-s − 1.04i·17^-s − 6.43i·19^-s + 5.28i·22^-s − 7.47·23^-s − 2.19·26^-s + (−2.64 − 0.0951i)28^-s − 7.47i·29^-s + ⋯

L(s) = 1

+ 0.707·2^-s + 0.5·4^-s + (−0.999 − 0.0359i)7^-s + 0.353·8^-s + 1.59i·11^-s − 0.607·13^-s + (−0.706 − 0.0254i)14^-s + 0.250·16^-s − 0.253i·17^-s − 1.47i·19^-s + 1.12i·22^-s − 1.55·23^-s − 0.429·26^-s + (−0.499 − 0.0179i)28^-s − 1.38i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3150$ = $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$
Sign:	$-0.503 + 0.863i$
Analytic conductor:	$25.1528$
Root analytic conductor:	$5.01526$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3150} (3149, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3150,\ (\ :1/2),\ -0.503 + 0.863i)$

Particular Values

$L(1)$	$\approx$	$1.086077824$
$L(\frac12)$	$\approx$	$1.086077824$
$L(\frac{3}{2})$		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 - T $
	3	$ 1 $
	5	$ 1 $
	7	$ 1 + (2.64 + 0.0951i)T $
good	11	$ 1 - 5.28iT - 11T^{2} $
	13	$ 1 + 2.19T + 13T^{2} $
	17	$ 1 + 1.04iT - 17T^{2} $
	19	$ 1 + 6.43iT - 19T^{2} $
	23	$ 1 + 7.47T + 23T^{2} $
	29	$ 1 + 7.47iT - 29T^{2} $
	31	$ 1 + 9.09iT - 31T^{2} $
	37	$ 1 - 0.855iT - 37T^{2} $
	41	$ 1 + 2.19T + 41T^{2} $
	43	$ 1 + 0.954iT - 43T^{2} $
	47	$ 1 + 11.0iT - 47T^{2} $
	53	$ 1 - 3.09T + 53T^{2} $
	59	$ 1 - 13.7T + 59T^{2} $
	61	$ 1 - 8.05iT - 61T^{2} $
	67	$ 1 + 5.33iT - 67T^{2} $
	71	$ 1 + 6.43iT - 71T^{2} $
	73	$ 1 + 4.57T + 73T^{2} $
	79	$ 1 + 15.6T + 79T^{2} $
	83	$ 1 - 4.38iT - 83T^{2} $
	89	$ 1 - 4.28T + 89T^{2} $
	97	$ 1 + 11.8T + 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

−8.322083192810818460354717705106, −7.32792033151827832735832527101, −6.99277901574765661101529115608, −6.14389298626220996784144724046, −5.34082685368333877435514141345, −4.39253173457436890138673287534, −3.93142246593743288893567514538, −2.60353331875382842459920240141, −2.15828539523710888270885430788, −0.24417949452123263842519559640, 1.40105246598310387289546053170, 2.73353709572955696369996338504, 3.44741772341814301699968409238, 4.04188764774939784795959450698, 5.30449143489844164978353212323, 5.86226434260004731243847136573, 6.47178123312405743660240213412, 7.27822501348135362951924994664, 8.234245956610141589294581188966, 8.766421612219231158240890194025

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.086077824\)
\(L(\frac12)\)	\(\approx\)	\(1.086077824\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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