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Properties

Label 2-930-31.8-c1-0-21
Degree $2$
Conductor $930$
Sign $-0.986 + 0.166i$
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

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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 + (−4.03 − 2.92i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (3.07 + 2.23i)11-s + (−0.309 + 0.951i)12-s + (−0.484 − 1.49i)13-s + (4.03 − 2.92i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−4.20 + 3.05i)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 + (−1.52 − 1.10i)7-s + (0.286 − 0.207i)8-s + (−0.269 + 0.195i)9-s + (−0.0977 + 0.300i)10-s + (0.928 + 0.674i)11-s + (−0.0892 + 0.274i)12-s + (−0.134 − 0.413i)13-s + (1.07 − 0.782i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (−1.02 + 0.741i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.986 + 0.166i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.986 + 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00786564 - 0.0940321i\)
\(L(\frac12)\) \(\approx\) \(0.00786564 - 0.0940321i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.47 + 1.02i)T \)
good7 \( 1 + (4.03 + 2.92i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-3.07 - 2.23i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.484 + 1.49i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.20 - 3.05i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.24 + 3.83i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.76 - 2.00i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.32 - 4.08i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + (-1.39 + 4.30i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (2.13 - 6.58i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (0.665 + 2.04i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.26 - 6.00i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.59 + 4.90i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 8.45T + 61T^{2} \)
67 \( 1 + 1.93T + 67T^{2} \)
71 \( 1 + (-6.60 + 4.79i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-11.7 - 8.52i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (7.75 - 5.63i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.46 + 13.7i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (5.21 + 3.78i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.87 + 2.08i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.497605786526851667150241140888, −9.007188868272811473346588834327, −7.70741199259229656451200321672, −6.77635160484416753382657720725, −6.68744526953062932425622871088, −5.60169006854159631847265086538, −4.34380347655268016478675091730, −3.32192488762528966868801580738, −1.63376043380938485498843773089, −0.04693900898749707848617358058, 2.03848140814369709933870012991, 3.14453393284589450472443011985, 3.92324570823732620057060209382, 5.26357861084240656214006065806, 6.14058385219636090816648642574, 6.79798444574143908123981452937, 8.418793147948343350469117686818, 9.269629236886247363859196404995, 9.423341001632402816402622550509, 10.32454523714428472444395074905

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