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Properties

Label 2-1375-1375.186-c0-0-0
Degree $2$
Conductor $1375$
Sign $-0.711 + 0.702i$
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  + (−1.92 + 0.493i)3-s + (−0.929 − 0.368i)4-s + (0.309 − 0.951i)5-s + (2.57 − 1.41i)9-s + (−0.187 + 0.982i)11-s + (1.96 + 0.248i)12-s + (−0.124 + 1.98i)15-s + (0.728 + 0.684i)16-s + (−0.637 + 0.770i)20-s + (−0.824 − 1.75i)23-s + (−0.809 − 0.587i)25-s + (−2.80 + 2.63i)27-s + (0.791 − 0.313i)31-s + (−0.124 − 1.98i)33-s + (−2.91 + 0.368i)36-s + (−1.17 − 1.10i)37-s + ⋯
L(s)  = 1  + (−1.92 + 0.493i)3-s + (−0.929 − 0.368i)4-s + (0.309 − 0.951i)5-s + (2.57 − 1.41i)9-s + (−0.187 + 0.982i)11-s + (1.96 + 0.248i)12-s + (−0.124 + 1.98i)15-s + (0.728 + 0.684i)16-s + (−0.637 + 0.770i)20-s + (−0.824 − 1.75i)23-s + (−0.809 − 0.587i)25-s + (−2.80 + 2.63i)27-s + (0.791 − 0.313i)31-s + (−0.124 − 1.98i)33-s + (−2.91 + 0.368i)36-s + (−1.17 − 1.10i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.711 + 0.702i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ -0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2397343693\)
\(L(\frac12)\) \(\approx\) \(0.2397343693\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.187 - 0.982i)T \)
good2 \( 1 + (0.929 + 0.368i)T^{2} \)
3 \( 1 + (1.92 - 0.493i)T + (0.876 - 0.481i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.535 + 0.844i)T^{2} \)
17 \( 1 + (-0.728 + 0.684i)T^{2} \)
19 \( 1 + (-0.876 - 0.481i)T^{2} \)
23 \( 1 + (0.824 + 1.75i)T + (-0.637 + 0.770i)T^{2} \)
29 \( 1 + (0.187 + 0.982i)T^{2} \)
31 \( 1 + (-0.791 + 0.313i)T + (0.728 - 0.684i)T^{2} \)
37 \( 1 + (1.17 + 1.10i)T + (0.0627 + 0.998i)T^{2} \)
41 \( 1 + (0.637 + 0.770i)T^{2} \)
43 \( 1 + (-0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.331 - 0.521i)T + (-0.425 + 0.904i)T^{2} \)
53 \( 1 + (0.0235 - 0.374i)T + (-0.992 - 0.125i)T^{2} \)
59 \( 1 + (1.44 + 0.182i)T + (0.968 + 0.248i)T^{2} \)
61 \( 1 + (0.637 - 0.770i)T^{2} \)
67 \( 1 + (0.0800 + 0.0967i)T + (-0.187 + 0.982i)T^{2} \)
71 \( 1 + (1.06 + 1.67i)T + (-0.425 + 0.904i)T^{2} \)
73 \( 1 + (-0.968 + 0.248i)T^{2} \)
79 \( 1 + (-0.876 + 0.481i)T^{2} \)
83 \( 1 + (-0.876 - 0.481i)T^{2} \)
89 \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \)
97 \( 1 + (0.929 - 1.12i)T + (-0.187 - 0.982i)T^{2} \)
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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

−9.660430964387821510409934258629, −9.020479645377092290278251111379, −7.88000212646841310817033855248, −6.63491875186063850989661022631, −5.93697515351379767221339865093, −5.19068836380407253702337915813, −4.56445165839706116648284626575, −4.12808279307673153330039117974, −1.61731323920266975435507739687, −0.28609424859910201703028926055, 1.45301004557504448097286585518, 3.23205697836272392602306001585, 4.32994954434382599534667363659, 5.39091794921348682767394530696, 5.81774308343655974234542177615, 6.69384049805473557832477015870, 7.47199550662712849827370449584, 8.255639865315064885679792105592, 9.630709142374323853180395566018, 10.19288905917436659552054292964

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