LMFDB - L-function 2-3100-31.30-c2-0-24

Dirichlet series

L(s) = 1

− 3.71i·3^-s − 1.89·7^-s − 4.78·9^-s − 3.40i·11^-s − 11.5i·13^-s + 23.9i·17^-s + 5.61·19^-s + 7.03i·21^-s − 5.50i·23^-s − 15.6i·27^-s + 50.5i·29^-s + (−30.5 + 5.11i)31^-s − 12.6·33^-s + 58.2i·37^-s − 42.9·39^-s + ⋯

L(s) = 1

− 1.23i·3^-s − 0.270·7^-s − 0.531·9^-s − 0.309i·11^-s − 0.890i·13^-s + 1.40i·17^-s + 0.295·19^-s + 0.335i·21^-s − 0.239i·23^-s − 0.579i·27^-s + 1.74i·29^-s + (−0.986 + 0.164i)31^-s − 0.383·33^-s + 1.57i·37^-s − 1.10·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$0.986 - 0.164i$
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:	$(2,\ 3100,\ (\ :1),\ 0.986 - 0.164i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.499735734$
$L(\frac12)$	$\approx$	$1.499735734$
$L(2)$		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 + (30.5 - 5.11i)T $
good	3	$ 1 + 3.71iT - 9T^{2} $
	7	$ 1 + 1.89T + 49T^{2} $
	11	$ 1 + 3.40iT - 121T^{2} $
	13	$ 1 + 11.5iT - 169T^{2} $
	17	$ 1 - 23.9iT - 289T^{2} $
	19	$ 1 - 5.61T + 361T^{2} $
	23	$ 1 + 5.50iT - 529T^{2} $
	29	$ 1 - 50.5iT - 841T^{2} $
	37	$ 1 - 58.2iT - 1.36e3T^{2} $
	41	$ 1 - 35.1T + 1.68e3T^{2} $
	43	$ 1 - 17.2iT - 1.84e3T^{2} $
	47	$ 1 - 69.6T + 2.20e3T^{2} $
	53	$ 1 + 14.5iT - 2.80e3T^{2} $
	59	$ 1 - 45.0T + 3.48e3T^{2} $
	61	$ 1 - 89.1iT - 3.72e3T^{2} $
	67	$ 1 + 129.T + 4.48e3T^{2} $
	71	$ 1 + 115.T + 5.04e3T^{2} $
	73	$ 1 + 97.3iT - 5.32e3T^{2} $
	79	$ 1 - 50.8iT - 6.24e3T^{2} $
	83	$ 1 + 59.9iT - 6.88e3T^{2} $
	89	$ 1 - 137. iT - 7.92e3T^{2} $
	97	$ 1 - 139.T + 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

−8.475984490414832514662313436818, −7.63666356212955160303302694456, −7.18434874762024754183311253121, −6.23215186239101240279588602174, −5.83466687887834031099843759583, −4.76426449059039422605177051961, −3.60093535993300399197155900409, −2.82705761590463318944906447007, −1.68758618861298098282274045605, −0.946710638065813855612519171501, 0.38834463642240140872678082891, 1.99487227599947390255699823864, 2.99990589271276458366787681772, 4.00387244792950501112908394728, 4.43986177982819550782904272059, 5.33514520291692570518842596323, 6.05351342801984386055156103689, 7.18782671666870604830791326551, 7.57905371992680000474552018501, 8.962549196911842104070468414691

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.499735734\)
\(L(\frac12)\)	\(\approx\)	\(1.499735734\)
\(L(2)\)		not available
\(L(1)\)		not available

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