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Properties

Label 2-1375-1375.21-c0-0-0
Degree $2$
Conductor $1375$
Sign $0.910 - 0.414i$
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  + (0.238 + 1.25i)3-s + (0.876 − 0.481i)4-s + (−0.809 − 0.587i)5-s + (−0.581 + 0.230i)9-s + (0.968 − 0.248i)11-s + (0.812 + 0.982i)12-s + (0.542 − 1.15i)15-s + (0.535 − 0.844i)16-s + (−0.992 − 0.125i)20-s + (−0.0235 + 0.374i)23-s + (0.309 + 0.951i)25-s + (0.255 + 0.403i)27-s + (0.110 + 0.0604i)31-s + (0.542 + 1.15i)33-s + (−0.398 + 0.481i)36-s + (0.331 − 0.521i)37-s + ⋯
L(s)  = 1  + (0.238 + 1.25i)3-s + (0.876 − 0.481i)4-s + (−0.809 − 0.587i)5-s + (−0.581 + 0.230i)9-s + (0.968 − 0.248i)11-s + (0.812 + 0.982i)12-s + (0.542 − 1.15i)15-s + (0.535 − 0.844i)16-s + (−0.992 − 0.125i)20-s + (−0.0235 + 0.374i)23-s + (0.309 + 0.951i)25-s + (0.255 + 0.403i)27-s + (0.110 + 0.0604i)31-s + (0.542 + 1.15i)33-s + (−0.398 + 0.481i)36-s + (0.331 − 0.521i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.910 - 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.332771780\)
\(L(\frac12)\) \(\approx\) \(1.332771780\)
\(L(1)\) 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 \( 1 + (-0.968 + 0.248i)T \)
good2 \( 1 + (-0.876 + 0.481i)T^{2} \)
3 \( 1 + (-0.238 - 1.25i)T + (-0.929 + 0.368i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.728 + 0.684i)T^{2} \)
17 \( 1 + (-0.535 - 0.844i)T^{2} \)
19 \( 1 + (0.929 + 0.368i)T^{2} \)
23 \( 1 + (0.0235 - 0.374i)T + (-0.992 - 0.125i)T^{2} \)
29 \( 1 + (-0.968 - 0.248i)T^{2} \)
31 \( 1 + (-0.110 - 0.0604i)T + (0.535 + 0.844i)T^{2} \)
37 \( 1 + (-0.331 + 0.521i)T + (-0.425 - 0.904i)T^{2} \)
41 \( 1 + (0.992 - 0.125i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (1.17 + 1.10i)T + (0.0627 + 0.998i)T^{2} \)
53 \( 1 + (0.824 - 1.75i)T + (-0.637 - 0.770i)T^{2} \)
59 \( 1 + (0.683 + 0.825i)T + (-0.187 + 0.982i)T^{2} \)
61 \( 1 + (0.992 + 0.125i)T^{2} \)
67 \( 1 + (-0.844 + 0.106i)T + (0.968 - 0.248i)T^{2} \)
71 \( 1 + (0.929 + 0.872i)T + (0.0627 + 0.998i)T^{2} \)
73 \( 1 + (0.187 + 0.982i)T^{2} \)
79 \( 1 + (0.929 - 0.368i)T^{2} \)
83 \( 1 + (0.929 + 0.368i)T^{2} \)
89 \( 1 + (0.929 - 1.12i)T + (-0.187 - 0.982i)T^{2} \)
97 \( 1 + (1.06 + 0.134i)T + (0.968 + 0.248i)T^{2} \)
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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

−9.674672488509662250202506630982, −9.272946884573233119940358705964, −8.353825103919285186515091564330, −7.44746954537534184928960137605, −6.53421145916966520340476378928, −5.53983121195141813009192523710, −4.64923011680614854826472141569, −3.85851779901414736673386311734, −3.02993809779307383045224208495, −1.40430755150021376517611223190, 1.46247967798656412897395712023, 2.51842973315462433766319795835, 3.40287027581572162325110831083, 4.41667185859445994565288023908, 6.09125212341443570621626881985, 6.79245549841795452945208159654, 7.12354375794351490527909908395, 8.025086453629120082701990139070, 8.433160695048409761072489825444, 9.748432689929586997354862648844

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