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Properties

Label 2-1375-1375.934-c0-0-0
Degree $2$
Conductor $1375$
Sign $-0.988 - 0.150i$
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  + (0.723 + 1.82i)3-s + (−0.535 + 0.844i)4-s + (−0.309 − 0.951i)5-s + (−2.08 + 1.95i)9-s + (−0.876 + 0.481i)11-s + (−1.92 − 0.368i)12-s + (1.51 − 1.25i)15-s + (−0.425 − 0.904i)16-s + (0.968 + 0.248i)20-s + (−0.0922 + 0.730i)23-s + (−0.809 + 0.587i)25-s + (−3.30 − 1.55i)27-s + (1.06 + 1.67i)31-s + (−1.51 − 1.25i)33-s + (−0.535 − 2.80i)36-s + (−1.06 + 0.500i)37-s + ⋯
L(s)  = 1  + (0.723 + 1.82i)3-s + (−0.535 + 0.844i)4-s + (−0.309 − 0.951i)5-s + (−2.08 + 1.95i)9-s + (−0.876 + 0.481i)11-s + (−1.92 − 0.368i)12-s + (1.51 − 1.25i)15-s + (−0.425 − 0.904i)16-s + (0.968 + 0.248i)20-s + (−0.0922 + 0.730i)23-s + (−0.809 + 0.587i)25-s + (−3.30 − 1.55i)27-s + (1.06 + 1.67i)31-s + (−1.51 − 1.25i)33-s + (−0.535 − 2.80i)36-s + (−1.06 + 0.500i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (934, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ -0.988 - 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8666683661\)
\(L(\frac12)\) \(\approx\) \(0.8666683661\)
\(L(1)\) 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 \( 1 + (0.876 - 0.481i)T \)
good2 \( 1 + (0.535 - 0.844i)T^{2} \)
3 \( 1 + (-0.723 - 1.82i)T + (-0.728 + 0.684i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.0627 - 0.998i)T^{2} \)
17 \( 1 + (-0.425 + 0.904i)T^{2} \)
19 \( 1 + (-0.728 - 0.684i)T^{2} \)
23 \( 1 + (0.0922 - 0.730i)T + (-0.968 - 0.248i)T^{2} \)
29 \( 1 + (-0.876 - 0.481i)T^{2} \)
31 \( 1 + (-1.06 - 1.67i)T + (-0.425 + 0.904i)T^{2} \)
37 \( 1 + (1.06 - 0.500i)T + (0.637 - 0.770i)T^{2} \)
41 \( 1 + (-0.968 + 0.248i)T^{2} \)
43 \( 1 + (0.309 - 0.951i)T^{2} \)
47 \( 1 + (-1.89 + 0.119i)T + (0.992 - 0.125i)T^{2} \)
53 \( 1 + (-0.742 + 0.614i)T + (0.187 - 0.982i)T^{2} \)
59 \( 1 + (-0.159 + 0.836i)T + (-0.929 - 0.368i)T^{2} \)
61 \( 1 + (-0.968 - 0.248i)T^{2} \)
67 \( 1 + (-0.383 - 1.49i)T + (-0.876 + 0.481i)T^{2} \)
71 \( 1 + (0.0235 + 0.374i)T + (-0.992 + 0.125i)T^{2} \)
73 \( 1 + (-0.929 + 0.368i)T^{2} \)
79 \( 1 + (-0.728 + 0.684i)T^{2} \)
83 \( 1 + (0.728 + 0.684i)T^{2} \)
89 \( 1 + (-0.0235 - 0.123i)T + (-0.929 + 0.368i)T^{2} \)
97 \( 1 + (0.450 - 1.75i)T + (-0.876 - 0.481i)T^{2} \)
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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

−10.00602426065464518809307579345, −9.251907766851190975570699357468, −8.607851413513497483342743105596, −8.211730500596657259785178732953, −7.32399750552737673281555946367, −5.32565507305299117812322896508, −5.01330902432121698570211662855, −4.13858825151565378376058630508, −3.51756475596372741862730276106, −2.51561705112159601757447527049, 0.65010933790305929851375691188, 2.15697220720568276998716105515, 2.82476586466569625186665371889, 4.05382752590797226561747858411, 5.68896755154832678924537760876, 6.16313179033479300236132620003, 7.05442856065017533722096504160, 7.72365649040998225330003327944, 8.435400053819591275961585980552, 9.151688292748279452734424514105

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