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

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3150$ = $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$
Sign:	$-0.982 + 0.186i$
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:	$(2,\ 3150,\ (\ :1/2),\ -0.982 + 0.186i)$

Particular Values

$L(1)$	$\approx$	$0.2167560032$
$L(\frac12)$	$\approx$	$0.2167560032$
$L(\frac{3}{2})$		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

−9.240556768624568506878709091006, −8.384279543524990020740803319890, −7.44659368747044618104563796266, −7.04025825690318944914603326019, −6.29517152787321049817042645667, −5.32792405072021427235231321450, −4.43798177089736015425056891710, −3.36932137167706376564718741870, −2.48922845705925419193822425508, −1.47288865494597226984905351152, 0.094314188738766299298412955579, 1.12224560123391708437908720527, 2.84612159361059767283120751754, 3.01686338833618637410252021198, 4.33016766888188145902847476431, 5.43532301720846821384179903633, 6.15701821214629007723426480861, 6.88505079597123975528613960797, 7.50500047182248762309500269887, 8.456246989889131524581580386160

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2167560032\)
\(L(\frac12)\)	\(\approx\)	\(0.2167560032\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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