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Properties

Label 2-1375-1375.219-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  + (−1.52 + 0.718i)3-s + (0.992 − 0.125i)4-s + (0.809 − 0.587i)5-s + (1.18 − 1.42i)9-s + (−0.0627 − 0.998i)11-s + (−1.42 + 0.904i)12-s + (−0.813 + 1.47i)15-s + (0.968 − 0.248i)16-s + (0.728 − 0.684i)20-s + (−0.666 − 1.68i)23-s + (0.309 − 0.951i)25-s + (−0.357 + 1.39i)27-s + (−1.84 − 0.233i)31-s + (0.813 + 1.47i)33-s + (0.992 − 1.56i)36-s + (0.473 + 1.84i)37-s + ⋯
L(s)  = 1  + (−1.52 + 0.718i)3-s + (0.992 − 0.125i)4-s + (0.809 − 0.587i)5-s + (1.18 − 1.42i)9-s + (−0.0627 − 0.998i)11-s + (−1.42 + 0.904i)12-s + (−0.813 + 1.47i)15-s + (0.968 − 0.248i)16-s + (0.728 − 0.684i)20-s + (−0.666 − 1.68i)23-s + (0.309 − 0.951i)25-s + (−0.357 + 1.39i)27-s + (−1.84 − 0.233i)31-s + (0.813 + 1.47i)33-s + (0.992 − 1.56i)36-s + (0.473 + 1.84i)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: $\chi_{1375} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.910 + 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9433773275\)
\(L(\frac12)\) \(\approx\) \(0.9433773275\)
\(L(1)\) 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 \( 1 + (0.0627 + 0.998i)T \)
good2 \( 1 + (-0.992 + 0.125i)T^{2} \)
3 \( 1 + (1.52 - 0.718i)T + (0.637 - 0.770i)T^{2} \)
7 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.187 - 0.982i)T^{2} \)
17 \( 1 + (0.968 + 0.248i)T^{2} \)
19 \( 1 + (0.637 + 0.770i)T^{2} \)
23 \( 1 + (0.666 + 1.68i)T + (-0.728 + 0.684i)T^{2} \)
29 \( 1 + (-0.0627 + 0.998i)T^{2} \)
31 \( 1 + (1.84 + 0.233i)T + (0.968 + 0.248i)T^{2} \)
37 \( 1 + (-0.473 - 1.84i)T + (-0.876 + 0.481i)T^{2} \)
41 \( 1 + (-0.728 - 0.684i)T^{2} \)
43 \( 1 + (-0.809 - 0.587i)T^{2} \)
47 \( 1 + (-1.15 - 0.220i)T + (0.929 + 0.368i)T^{2} \)
53 \( 1 + (-0.961 + 1.74i)T + (-0.535 - 0.844i)T^{2} \)
59 \( 1 + (-1.03 - 1.63i)T + (-0.425 + 0.904i)T^{2} \)
61 \( 1 + (-0.728 + 0.684i)T^{2} \)
67 \( 1 + (-0.659 + 0.702i)T + (-0.0627 - 0.998i)T^{2} \)
71 \( 1 + (0.200 - 1.05i)T + (-0.929 - 0.368i)T^{2} \)
73 \( 1 + (-0.425 - 0.904i)T^{2} \)
79 \( 1 + (0.637 - 0.770i)T^{2} \)
83 \( 1 + (-0.637 - 0.770i)T^{2} \)
89 \( 1 + (-0.200 + 0.316i)T + (-0.425 - 0.904i)T^{2} \)
97 \( 1 + (-0.340 - 0.362i)T + (-0.0627 + 0.998i)T^{2} \)
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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

−10.14806419439151196074098417177, −9.117194633500450442769009348885, −8.168206019012615835241020173876, −6.85692463172516448177453706519, −6.18347695423616104466647938620, −5.68987087251713100969412880115, −4.96902309006434805207951958675, −3.89104954471689589972526725760, −2.44743203140598708789461453513, −0.998537371765420855711265507666, 1.60944277667116818686008988413, 2.24707115476144043199740900202, 3.79529026007733658710881015088, 5.44174156375630463962090466335, 5.65000529090968275677470986069, 6.59565139301754611793545915744, 7.26782034671588212909561884826, 7.57422153853006521130173450362, 9.293947816462096211704963412321, 10.13478364792995372396276441584

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