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Properties

Label 2-310-155.109-c1-0-5
Degree $2$
Conductor $310$
Sign $0.959 + 0.280i$
Analytic cond. $2.47536$
Root an. cond. $1.57332$
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  + (−0.587 − 0.809i)2-s + (0.165 − 0.227i)3-s + (−0.309 + 0.951i)4-s + (−2.22 − 0.192i)5-s − 0.281·6-s + (3.62 + 1.17i)7-s + (0.951 − 0.309i)8-s + (0.902 + 2.77i)9-s + (1.15 + 1.91i)10-s + (−0.493 + 1.51i)11-s + (0.165 + 0.227i)12-s + (1.88 − 2.59i)13-s + (−1.17 − 3.62i)14-s + (−0.412 + 0.475i)15-s + (−0.809 − 0.587i)16-s + (5.38 − 1.74i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (0.0955 − 0.131i)3-s + (−0.154 + 0.475i)4-s + (−0.996 − 0.0862i)5-s − 0.114·6-s + (1.36 + 0.444i)7-s + (0.336 − 0.109i)8-s + (0.300 + 0.925i)9-s + (0.364 + 0.605i)10-s + (−0.148 + 0.458i)11-s + (0.0477 + 0.0657i)12-s + (0.523 − 0.720i)13-s + (−0.314 − 0.968i)14-s + (−0.106 + 0.122i)15-s + (−0.202 − 0.146i)16-s + (1.30 − 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $0.959 + 0.280i$
Analytic conductor: \(2.47536\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1/2),\ 0.959 + 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08652 - 0.155546i\)
\(L(\frac12)\) \(\approx\) \(1.08652 - 0.155546i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (2.22 + 0.192i)T \)
31 \( 1 + (-4.81 - 2.80i)T \)
good3 \( 1 + (-0.165 + 0.227i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-3.62 - 1.17i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.493 - 1.51i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.88 + 2.59i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.38 + 1.74i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.03 + 1.47i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.41 - 1.43i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.33 + 0.969i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 1.74iT - 37T^{2} \)
41 \( 1 + (7.29 - 5.30i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-0.0681 - 0.0937i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-5.45 + 7.51i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.57 - 1.81i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.25 - 5.27i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 + 0.561iT - 67T^{2} \)
71 \( 1 + (-0.785 - 2.41i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.99 + 1.29i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.88 + 8.88i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (9.27 + 12.7i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.58 - 14.1i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.42 + 1.76i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.75608904832434018633770969694, −10.77346998681344449860723074404, −9.992541271347141055324350883233, −8.504050996723507415343521714683, −8.003806761213877669332407812917, −7.35285593344252112783022673665, −5.33589793400454754036968184323, −4.47590026936746332122559487707, −2.99665357923575571231940956096, −1.44450912466096286892429676658, 1.18639694488516160109192114010, 3.64267579670079372877128110989, 4.54620663659694989995717137218, 5.92392738404114507514548822205, 7.12587391536141163830945080678, 8.018309557645384301973078857625, 8.540543857095230839346551797588, 9.816395385440402065847593683084, 10.79387052766633871403501548731, 11.64966443727665906550643053255

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