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Properties

Label 2-1375-1375.1154-c0-0-0
Degree $2$
Conductor $1375$
Sign $0.675 + 0.737i$
Analytic cond. $0.686214$
Root an. cond. $0.828380$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.239 − 0.435i)3-s + (−0.728 − 0.684i)4-s + (0.809 + 0.587i)5-s + (0.403 − 0.635i)9-s + (0.929 + 0.368i)11-s + (−0.123 + 0.481i)12-s + (0.0623 − 0.493i)15-s + (0.0627 + 0.998i)16-s + (−0.187 − 0.982i)20-s + (−0.742 − 0.614i)23-s + (0.309 + 0.951i)25-s + (−0.869 − 0.0547i)27-s + (0.929 − 0.872i)31-s + (−0.0623 − 0.493i)33-s + (−0.728 + 0.187i)36-s + (1.89 − 0.119i)37-s + ⋯
L(s)  = 1  + (−0.239 − 0.435i)3-s + (−0.728 − 0.684i)4-s + (0.809 + 0.587i)5-s + (0.403 − 0.635i)9-s + (0.929 + 0.368i)11-s + (−0.123 + 0.481i)12-s + (0.0623 − 0.493i)15-s + (0.0627 + 0.998i)16-s + (−0.187 − 0.982i)20-s + (−0.742 − 0.614i)23-s + (0.309 + 0.951i)25-s + (−0.869 − 0.0547i)27-s + (0.929 − 0.872i)31-s + (−0.0623 − 0.493i)33-s + (−0.728 + 0.187i)36-s + (1.89 − 0.119i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.675 + 0.737i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.675 + 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.049562633\)
\(L(\frac12)\) \(\approx\) \(1.049562633\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.929 - 0.368i)T \)
good2 \( 1 + (0.728 + 0.684i)T^{2} \)
3 \( 1 + (0.239 + 0.435i)T + (-0.535 + 0.844i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.425 - 0.904i)T^{2} \)
17 \( 1 + (0.0627 - 0.998i)T^{2} \)
19 \( 1 + (-0.535 - 0.844i)T^{2} \)
23 \( 1 + (0.742 + 0.614i)T + (0.187 + 0.982i)T^{2} \)
29 \( 1 + (0.929 - 0.368i)T^{2} \)
31 \( 1 + (-0.929 + 0.872i)T + (0.0627 - 0.998i)T^{2} \)
37 \( 1 + (-1.89 + 0.119i)T + (0.992 - 0.125i)T^{2} \)
41 \( 1 + (0.187 - 0.982i)T^{2} \)
43 \( 1 + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (1.06 + 0.500i)T + (0.637 + 0.770i)T^{2} \)
53 \( 1 + (-0.0922 + 0.730i)T + (-0.968 - 0.248i)T^{2} \)
59 \( 1 + (-0.121 - 0.0312i)T + (0.876 + 0.481i)T^{2} \)
61 \( 1 + (0.187 + 0.982i)T^{2} \)
67 \( 1 + (0.246 + 0.0469i)T + (0.929 + 0.368i)T^{2} \)
71 \( 1 + (0.824 - 1.75i)T + (-0.637 - 0.770i)T^{2} \)
73 \( 1 + (0.876 - 0.481i)T^{2} \)
79 \( 1 + (-0.535 + 0.844i)T^{2} \)
83 \( 1 + (0.535 + 0.844i)T^{2} \)
89 \( 1 + (-0.824 + 0.211i)T + (0.876 - 0.481i)T^{2} \)
97 \( 1 + (1.96 - 0.374i)T + (0.929 - 0.368i)T^{2} \)
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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

−9.825795716628586247303089321670, −9.117317063914794095652802415992, −8.112334608845114104871439946393, −6.92088218645310608919512726504, −6.34010061035905303771907784312, −5.77884608345039649775445173347, −4.59785421090353189774596015332, −3.75992628513845933109377180759, −2.22874527982273044802854840468, −1.13214874854336410026036614170, 1.43857664123639013649010175916, 2.92685802862503643006755980358, 4.17718018341781724059397360781, 4.66338310385346462966822513330, 5.61704492662047918444030734506, 6.46546295098860302429317170498, 7.70378588745161717981808729082, 8.332631277543724852717671523427, 9.302081199458061172005161739707, 9.619714149874198002177083758554

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