LMFDB - L-function 2-2312-17.16-c1-0-14

Dirichlet series

L(s) = 1

− 3.30i·3^-s + 2.30i·5^-s − 2.30i·7^-s − 7.90·9^-s + 5.60i·11^-s + 0.697·13^-s + 7.60·15^-s + 1.30·19^-s − 7.60·21^-s + 4.30i·23^-s − 0.302·25^-s + 16.2i·27^-s + 0.697i·29^-s + 4.21i·31^-s + 18.5·33^-s + ⋯

L(s) = 1

− 1.90i·3^-s + 1.02i·5^-s − 0.870i·7^-s − 2.63·9^-s + 1.69i·11^-s + 0.193·13^-s + 1.96·15^-s + 0.298·19^-s − 1.65·21^-s + 0.897i·23^-s − 0.0605·25^-s + 3.11i·27^-s + 0.129i·29^-s + 0.756i·31^-s + 3.22·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2312$ = $2^{3} \cdot 17^{2}$
Sign:	$0.970 + 0.242i$
Analytic conductor:	$18.4614$
Root analytic conductor:	$4.29667$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2312} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2312,\ (\ :1/2),\ 0.970 + 0.242i)$

Particular Values

$L(1)$	$\approx$	$1.467906074$
$L(\frac12)$	$\approx$	$1.467906074$
$L(\frac{3}{2})$		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 $
	17	$ 1 $
good	3	$ 1 + 3.30iT - 3T^{2} $
	5	$ 1 - 2.30iT - 5T^{2} $
	7	$ 1 + 2.30iT - 7T^{2} $
	11	$ 1 - 5.60iT - 11T^{2} $
	13	$ 1 - 0.697T + 13T^{2} $
	19	$ 1 - 1.30T + 19T^{2} $
	23	$ 1 - 4.30iT - 23T^{2} $
	29	$ 1 - 0.697iT - 29T^{2} $
	31	$ 1 - 4.21iT - 31T^{2} $
	37	$ 1 + 8.60iT - 37T^{2} $
	41	$ 1 - 6iT - 41T^{2} $
	43	$ 1 - 4.21T + 43T^{2} $
	47	$ 1 - 11.6T + 47T^{2} $
	53	$ 1 + 3.30T + 53T^{2} $
	59	$ 1 + 3.21T + 59T^{2} $
	61	$ 1 + 1.60iT - 61T^{2} $
	67	$ 1 - 0.605T + 67T^{2} $
	71	$ 1 - 12.4iT - 71T^{2} $
	73	$ 1 + 2.39iT - 73T^{2} $
	79	$ 1 - 10iT - 79T^{2} $
	83	$ 1 - 8.69T + 83T^{2} $
	89	$ 1 - 7.21T + 89T^{2} $
	97	$ 1 + 16.1iT - 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

−8.805579703893068809005935410142, −7.60470546003141330892266964644, −7.43719656515633025999766787891, −6.90807589606569252411270563691, −6.22441707524668959483118320465, −5.24095940842544777602590076762, −3.92988223487238696565908553261, −2.84000740622992351406648414370, −2.03367924361778573162389548298, −1.07131784938734152739301438612, 0.58626506011485960290255830520, 2.60238798617430226828573007818, 3.41699535261858076039805737786, 4.27972530416725569915346643006, 5.01887540151646959678273090716, 5.68797707316617382473419829808, 6.17448680969136112878166992023, 8.000979431456172374631743801332, 8.698487914722048333730989189283, 8.937405276973575988323883765414

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

\(L(1)\)	\(\approx\)	\(1.467906074\)
\(L(\frac12)\)	\(\approx\)	\(1.467906074\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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