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Properties

Label 2-145-29.28-c1-0-2
Degree $2$
Conductor $145$
Sign $0.356 - 0.934i$
Analytic cond. $1.15783$
Root an. cond. $1.07602$
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  − 0.285i·2-s + 3.21i·3-s + 1.91·4-s + 5-s + 0.918·6-s − 2.91·7-s − 1.11i·8-s − 7.35·9-s − 0.285i·10-s − 3.21i·11-s + 6.17i·12-s + 4.35·13-s + 0.833i·14-s + 3.21i·15-s + 3.51·16-s + 1.97i·17-s + ⋯
L(s)  = 1  − 0.201i·2-s + 1.85i·3-s + 0.959·4-s + 0.447·5-s + 0.374·6-s − 1.10·7-s − 0.395i·8-s − 2.45·9-s − 0.0902i·10-s − 0.970i·11-s + 1.78i·12-s + 1.20·13-s + 0.222i·14-s + 0.830i·15-s + 0.879·16-s + 0.478i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.356 - 0.934i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ 0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05094 + 0.724041i\)
\(L(\frac12)\) \(\approx\) \(1.05094 + 0.724041i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
29 \( 1 + (-1.91 + 5.03i)T \)
good2 \( 1 + 0.285iT - 2T^{2} \)
3 \( 1 - 3.21iT - 3T^{2} \)
7 \( 1 + 2.91T + 7T^{2} \)
11 \( 1 + 3.21iT - 11T^{2} \)
13 \( 1 - 4.35T + 13T^{2} \)
17 \( 1 - 1.97iT - 17T^{2} \)
19 \( 1 + 1.40iT - 19T^{2} \)
23 \( 1 + 3.43T + 23T^{2} \)
31 \( 1 - 8.98iT - 31T^{2} \)
37 \( 1 + 4.46iT - 37T^{2} \)
41 \( 1 + 9.39iT - 41T^{2} \)
43 \( 1 - 6.84iT - 43T^{2} \)
47 \( 1 + 3.21iT - 47T^{2} \)
53 \( 1 + 6.19T + 53T^{2} \)
59 \( 1 - 2.35T + 59T^{2} \)
61 \( 1 - 8.24iT - 61T^{2} \)
67 \( 1 + 0.563T + 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 - 6.84iT - 79T^{2} \)
83 \( 1 + 1.08T + 83T^{2} \)
89 \( 1 - 3.62iT - 89T^{2} \)
97 \( 1 - 3.78iT - 97T^{2} \)
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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

−13.38749008424873166869801174055, −11.95099481558821663118668620233, −10.83164568131647930076743944234, −10.44094986287110729928438566500, −9.420842768474403813740335725819, −8.467120486958505743585647087868, −6.37008950890303246724737307192, −5.68841784476103431663595421947, −3.84152268153108219897886448728, −3.00175403977829758653362093676, 1.66033910003947287562029708493, 2.94643022032813965197579958589, 5.90084483572000745385938935506, 6.48201322985723351451996667291, 7.28968966621003681098976352443, 8.317313900846187687773881134870, 9.799645596829058490474614684826, 11.20311866281964457212622830379, 12.13566024639333329940522805515, 12.87911327953472097115144907683

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