LMFDB - L-function 2-3100-31.30-c2-0-37

Dirichlet series

L(s) = 1

− 3.53i·3^-s − 7.77·7^-s − 3.49·9^-s + 10.7i·11^-s − 10.7i·13^-s − 21.5i·17^-s + 18.1·19^-s + 27.4i·21^-s + 13.0i·23^-s − 19.4i·27^-s + 38.9i·29^-s + (26.0 + 16.7i)31^-s + 38.0·33^-s + 48.3i·37^-s − 37.8·39^-s + ⋯

L(s) = 1

− 1.17i·3^-s − 1.11·7^-s − 0.387·9^-s + 0.979i·11^-s − 0.824i·13^-s − 1.26i·17^-s + 0.954·19^-s + 1.30i·21^-s + 0.568i·23^-s − 0.721i·27^-s + 1.34i·29^-s + (0.841 + 0.540i)31^-s + 1.15·33^-s + 1.30i·37^-s − 0.970·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$0.841 + 0.540i$
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:	$(2,\ 3100,\ (\ :1),\ 0.841 + 0.540i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.608291812$
$L(\frac12)$	$\approx$	$1.608291812$
$L(2)$		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 + (-26.0 - 16.7i)T $
good	3	$ 1 + 3.53iT - 9T^{2} $
	7	$ 1 + 7.77T + 49T^{2} $
	11	$ 1 - 10.7iT - 121T^{2} $
	13	$ 1 + 10.7iT - 169T^{2} $
	17	$ 1 + 21.5iT - 289T^{2} $
	19	$ 1 - 18.1T + 361T^{2} $
	23	$ 1 - 13.0iT - 529T^{2} $
	29	$ 1 - 38.9iT - 841T^{2} $
	37	$ 1 - 48.3iT - 1.36e3T^{2} $
	41	$ 1 + 63.7T + 1.68e3T^{2} $
	43	$ 1 - 2.27iT - 1.84e3T^{2} $
	47	$ 1 + 72.8T + 2.20e3T^{2} $
	53	$ 1 - 100. iT - 2.80e3T^{2} $
	59	$ 1 - 117.T + 3.48e3T^{2} $
	61	$ 1 + 20.9iT - 3.72e3T^{2} $
	67	$ 1 - 15.3T + 4.48e3T^{2} $
	71	$ 1 - 61.1T + 5.04e3T^{2} $
	73	$ 1 + 40.9iT - 5.32e3T^{2} $
	79	$ 1 - 95.6iT - 6.24e3T^{2} $
	83	$ 1 + 28.0iT - 6.88e3T^{2} $
	89	$ 1 + 16.8iT - 7.92e3T^{2} $
	97	$ 1 - 171.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

−8.242352116719530290875593952808, −7.55324969425668054068370311911, −6.85184382550724890585914285336, −6.60366449599131424889408544650, −5.42665153799671028022955382206, −4.78125141767371061765411102377, −3.33666485688659647234169149597, −2.84063955867562631955889535701, −1.63080067659525151140184494989, −0.71994524711808822694216853862, 0.53028133422494789623272081034, 2.10529219269452426976301251985, 3.39922203508551278265750534402, 3.69495916361248507062510162911, 4.60103855212605670437600994934, 5.52958105607650719180521460618, 6.28462280476993386359322928345, 6.87849445965970561212806545707, 8.100661170429066588805980078954, 8.689865635277826544772947381272

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.608291812\)
\(L(\frac12)\)	\(\approx\)	\(1.608291812\)
\(L(2)\)		not available
\(L(1)\)		not available

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