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Properties

Label 2-1375-1375.384-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  + (−1.36 + 0.0859i)3-s + (0.637 + 0.770i)4-s + (−0.309 − 0.951i)5-s + (0.867 − 0.109i)9-s + (0.425 − 0.904i)11-s + (−0.937 − 0.998i)12-s + (0.503 + 1.27i)15-s + (−0.187 + 0.982i)16-s + (0.535 − 0.844i)20-s + (−0.961 − 1.74i)23-s + (−0.809 + 0.587i)25-s + (0.168 − 0.0322i)27-s + (1.11 − 1.35i)31-s + (−0.503 + 1.27i)33-s + (0.637 + 0.598i)36-s + (1.15 + 0.220i)37-s + ⋯
L(s)  = 1  + (−1.36 + 0.0859i)3-s + (0.637 + 0.770i)4-s + (−0.309 − 0.951i)5-s + (0.867 − 0.109i)9-s + (0.425 − 0.904i)11-s + (−0.937 − 0.998i)12-s + (0.503 + 1.27i)15-s + (−0.187 + 0.982i)16-s + (0.535 − 0.844i)20-s + (−0.961 − 1.74i)23-s + (−0.809 + 0.587i)25-s + (0.168 − 0.0322i)27-s + (1.11 − 1.35i)31-s + (−0.503 + 1.27i)33-s + (0.637 + 0.598i)36-s + (1.15 + 0.220i)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: $\chi_{1375} (384, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7069748485\)
\(L(\frac12)\) \(\approx\) \(0.7069748485\)
\(L(1)\) 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 \( 1 + (-0.425 + 0.904i)T \)
good2 \( 1 + (-0.637 - 0.770i)T^{2} \)
3 \( 1 + (1.36 - 0.0859i)T + (0.992 - 0.125i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.968 - 0.248i)T^{2} \)
17 \( 1 + (-0.187 - 0.982i)T^{2} \)
19 \( 1 + (0.992 + 0.125i)T^{2} \)
23 \( 1 + (0.961 + 1.74i)T + (-0.535 + 0.844i)T^{2} \)
29 \( 1 + (0.425 + 0.904i)T^{2} \)
31 \( 1 + (-1.11 + 1.35i)T + (-0.187 - 0.982i)T^{2} \)
37 \( 1 + (-1.15 - 0.220i)T + (0.929 + 0.368i)T^{2} \)
41 \( 1 + (-0.535 - 0.844i)T^{2} \)
43 \( 1 + (0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.473 + 1.84i)T + (-0.876 - 0.481i)T^{2} \)
53 \( 1 + (-0.666 - 1.68i)T + (-0.728 + 0.684i)T^{2} \)
59 \( 1 + (0.273 - 0.256i)T + (0.0627 - 0.998i)T^{2} \)
61 \( 1 + (-0.535 + 0.844i)T^{2} \)
67 \( 1 + (-0.621 + 0.394i)T + (0.425 - 0.904i)T^{2} \)
71 \( 1 + (-1.41 - 0.362i)T + (0.876 + 0.481i)T^{2} \)
73 \( 1 + (0.0627 + 0.998i)T^{2} \)
79 \( 1 + (0.992 - 0.125i)T^{2} \)
83 \( 1 + (-0.992 - 0.125i)T^{2} \)
89 \( 1 + (1.41 + 1.32i)T + (0.0627 + 0.998i)T^{2} \)
97 \( 1 + (1.65 + 1.05i)T + (0.425 + 0.904i)T^{2} \)
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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

−9.819363434309491425442502989477, −8.574145672857923585293454819519, −8.230674684744182750493856823969, −7.11326491520295851200100258336, −6.20296959177559917208806490557, −5.74463630171471718763667538928, −4.52650470759248656713983944419, −3.93448351652182149969379125507, −2.46063515059916612270954900464, −0.76528468281201838721013149177, 1.35781071947619600870961983824, 2.63117661706776421262132610219, 4.02276655503928816259125273787, 5.10622596625006454798502242285, 5.88763968887993626495281141220, 6.55928140416108814846692273402, 7.09424643937663595162733988795, 7.932777531498276314679999663111, 9.571831535426526242590540372712, 10.00214866556371873416593055051

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