↑

Properties

Label 2-930-31.4-c1-0-21
Degree $2$
Conductor $930$
Sign $-0.965 + 0.259i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s − 0.999·6-s + (−1.06 + 0.775i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (−0.567 + 0.412i)11-s + (0.309 + 0.951i)12-s + (0.501 − 1.54i)13-s + (1.06 + 0.775i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−4.24 − 3.08i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.178 − 0.549i)3-s + (−0.404 + 0.293i)4-s + 0.447·5-s − 0.408·6-s + (−0.403 + 0.293i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.195i)9-s + (−0.0977 − 0.300i)10-s + (−0.170 + 0.124i)11-s + (0.0892 + 0.274i)12-s + (0.139 − 0.427i)13-s + (0.285 + 0.207i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (−1.02 − 0.747i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.965 + 0.259i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.965 + 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129506 - 0.981984i\)
\(L(\frac12)\) \(\approx\) \(0.129506 - 0.981984i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 - T \)
31 \( 1 + (-4.55 + 3.20i)T \)
good7 \( 1 + (1.06 - 0.775i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.567 - 0.412i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.501 + 1.54i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.24 + 3.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.71 + 5.28i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.350 + 0.254i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.62 + 8.06i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 + (3.60 + 11.1i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.841 - 2.59i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-0.650 + 2.00i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.37 - 3.90i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.567 + 1.74i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 3.50T + 61T^{2} \)
67 \( 1 - 1.84T + 67T^{2} \)
71 \( 1 + (-0.612 - 0.445i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.635 - 0.461i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-8.87 - 6.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.02 - 6.22i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-4.32 + 3.14i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (10.7 - 7.80i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.575713649271958175516717868748, −9.025268018772969388232818767310, −8.178312481776472307831627382117, −7.15778204413609936662547354477, −6.33255519066521875961320008474, −5.27764999105674874024007606313, −4.15279072807109638610972918893, −2.77818164340780877627056217243, −2.17395415922432969077805703262, −0.47315144091856462591867351933, 1.74520675250748818091782825716, 3.33872315784734789879662748171, 4.31489158189081621483397818281, 5.30060410506893434923968312901, 6.28063562180372893205527833110, 6.89183738657484295979012832947, 8.130305595204003470014881860014, 8.733130169722296974218955181118, 9.550441874349665832096902496888, 10.34198921833032657270195635905

Graph of the $Z$-function along the critical line