Properties

Label 2-1375-1375.1066-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  + (0.348 − 0.137i)3-s + (0.535 − 0.844i)4-s + (0.309 + 0.951i)5-s + (−0.626 + 0.588i)9-s + (0.876 − 0.481i)11-s + (0.0702 − 0.368i)12-s + (0.238 + 0.288i)15-s + (−0.425 − 0.904i)16-s + (0.968 + 0.248i)20-s + (1.84 + 0.233i)23-s + (−0.809 + 0.587i)25-s + (−0.296 + 0.630i)27-s + (−1.06 − 1.67i)31-s + (0.238 − 0.288i)33-s + (0.161 + 0.844i)36-s + (0.688 + 1.46i)37-s + ⋯
L(s)  = 1  + (0.348 − 0.137i)3-s + (0.535 − 0.844i)4-s + (0.309 + 0.951i)5-s + (−0.626 + 0.588i)9-s + (0.876 − 0.481i)11-s + (0.0702 − 0.368i)12-s + (0.238 + 0.288i)15-s + (−0.425 − 0.904i)16-s + (0.968 + 0.248i)20-s + (1.84 + 0.233i)23-s + (−0.809 + 0.587i)25-s + (−0.296 + 0.630i)27-s + (−1.06 − 1.67i)31-s + (0.238 − 0.288i)33-s + (0.161 + 0.844i)36-s + (0.688 + 1.46i)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: $\chi_{1375} (1066, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.435167614\)
\(L(\frac12)\) \(\approx\) \(1.435167614\)
\(L(1)\) 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 \( 1 + (-0.348 + 0.137i)T + (0.728 - 0.684i)T^{2} \)
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 \( 1 + (-1.84 - 0.233i)T + (0.968 + 0.248i)T^{2} \)
29 \( 1 + (-0.876 - 0.481i)T^{2} \)
31 \( 1 + (1.06 + 1.67i)T + (-0.425 + 0.904i)T^{2} \)
37 \( 1 + (-0.688 - 1.46i)T + (-0.637 + 0.770i)T^{2} \)
41 \( 1 + (-0.968 + 0.248i)T^{2} \)
43 \( 1 + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (-0.0388 - 0.616i)T + (-0.992 + 0.125i)T^{2} \)
53 \( 1 + (1.11 + 1.35i)T + (-0.187 + 0.982i)T^{2} \)
59 \( 1 + (-0.159 + 0.836i)T + (-0.929 - 0.368i)T^{2} \)
61 \( 1 + (-0.968 - 0.248i)T^{2} \)
67 \( 1 + (1.23 - 0.317i)T + (0.876 - 0.481i)T^{2} \)
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 \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \)
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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

−9.660069335815935025392222887233, −9.202981874719485476734629644551, −8.046314869775991540754831587857, −7.21339590637989932253254989073, −6.44057779087443568833325492337, −5.82631661314716290250808884704, −4.87366628280298513786281417357, −3.36923559371122185973073482625, −2.61236435278439496055815234491, −1.53408678243934952776277110015, 1.50653561665538077210631884571, 2.78497824750466030228302951974, 3.68814299165976853798798977122, 4.59954289946479893004043332245, 5.67063426912688996027434976286, 6.65692067327776830410433213204, 7.35797345478071954870516970670, 8.404048952552510027870866011393, 9.026699974090399570028341975393, 9.362348453635640077860947291150

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