LMFDB - L-function 2-3100-31.30-c2-0-52

Dirichlet series

L(s) = 1

− 3.03i·3^-s + 12.6·7^-s − 0.187·9^-s + 17.2i·11^-s + 14.9i·13^-s − 2.62i·17^-s + 16.9·19^-s − 38.3i·21^-s + 0.547i·23^-s − 26.7i·27^-s − 33.2i·29^-s + (−30.1 + 7.11i)31^-s + 52.2·33^-s + 62.3i·37^-s + 45.3·39^-s + ⋯

L(s) = 1

− 1.01i·3^-s + 1.80·7^-s − 0.0208·9^-s + 1.56i·11^-s + 1.15i·13^-s − 0.154i·17^-s + 0.893·19^-s − 1.82i·21^-s + 0.0238i·23^-s − 0.989i·27^-s − 1.14i·29^-s + (−0.973 + 0.229i)31^-s + 1.58·33^-s + 1.68i·37^-s + 1.16·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$0.973 - 0.229i$
Analytic conductor:	$84.4688$
Root analytic conductor:	$9.19069$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3100} (1301, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3100,\ (\ :1),\ 0.973 - 0.229i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.884701392$
$L(\frac12)$	$\approx$	$2.884701392$
$L(2)$		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 $
	31	$ 1 + (30.1 - 7.11i)T $
good	3	$ 1 + 3.03iT - 9T^{2} $
	7	$ 1 - 12.6T + 49T^{2} $
	11	$ 1 - 17.2iT - 121T^{2} $
	13	$ 1 - 14.9iT - 169T^{2} $
	17	$ 1 + 2.62iT - 289T^{2} $
	19	$ 1 - 16.9T + 361T^{2} $
	23	$ 1 - 0.547iT - 529T^{2} $
	29	$ 1 + 33.2iT - 841T^{2} $
	37	$ 1 - 62.3iT - 1.36e3T^{2} $
	41	$ 1 + 10.8T + 1.68e3T^{2} $
	43	$ 1 - 13.8iT - 1.84e3T^{2} $
	47	$ 1 - 38.8T + 2.20e3T^{2} $
	53	$ 1 - 53.5iT - 2.80e3T^{2} $
	59	$ 1 + 44.2T + 3.48e3T^{2} $
	61	$ 1 + 71.3iT - 3.72e3T^{2} $
	67	$ 1 - 55.3T + 4.48e3T^{2} $
	71	$ 1 + 61.5T + 5.04e3T^{2} $
	73	$ 1 + 24.5iT - 5.32e3T^{2} $
	79	$ 1 - 130. iT - 6.24e3T^{2} $
	83	$ 1 - 122. iT - 6.88e3T^{2} $
	89	$ 1 - 149. iT - 7.92e3T^{2} $
	97	$ 1 + 25.3T + 9.40e3T^{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

−8.227178348284166495497656296381, −7.74722976056313369495625653820, −7.15400080528833814507622724499, −6.58371848615413401433232469254, −5.39619766343325952123363109240, −4.65525015547607791420673782441, −4.12276383264361504832379497725, −2.43543327821053138444286596109, −1.74820340694532614777546139593, −1.20416300851699776424941690169, 0.68754888688880808953112933805, 1.73233466415188648380599391878, 3.10666051911021905029494024343, 3.77101535973306819392908165216, 4.68991286489287914809545215807, 5.45076488012404370329698102504, 5.72811096847259797310488534116, 7.31437138311480574591023590114, 7.75930443488595909277791854588, 8.764488834494180202448682531062

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.884701392\)
\(L(\frac12)\)	\(\approx\)	\(2.884701392\)
\(L(2)\)		not available
\(L(1)\)		not available

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