Properties

Label 2-1375-1375.846-c0-0-0
Degree $2$
Conductor $1375$
Sign $0.675 + 0.737i$
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  + (1.69 − 0.933i)3-s + (0.728 + 0.684i)4-s + (−0.809 − 0.587i)5-s + (1.47 − 2.32i)9-s + (−0.929 − 0.368i)11-s + (1.87 + 0.481i)12-s + (−1.92 − 0.242i)15-s + (0.0627 + 0.998i)16-s + (−0.187 − 0.982i)20-s + (−1.11 + 1.35i)23-s + (0.309 + 0.951i)25-s + (0.213 − 3.38i)27-s + (−0.929 + 0.872i)31-s + (−1.92 + 0.242i)33-s + (2.66 − 0.684i)36-s + (0.0388 + 0.616i)37-s + ⋯
L(s)  = 1  + (1.69 − 0.933i)3-s + (0.728 + 0.684i)4-s + (−0.809 − 0.587i)5-s + (1.47 − 2.32i)9-s + (−0.929 − 0.368i)11-s + (1.87 + 0.481i)12-s + (−1.92 − 0.242i)15-s + (0.0627 + 0.998i)16-s + (−0.187 − 0.982i)20-s + (−1.11 + 1.35i)23-s + (0.309 + 0.951i)25-s + (0.213 − 3.38i)27-s + (−0.929 + 0.872i)31-s + (−1.92 + 0.242i)33-s + (2.66 − 0.684i)36-s + (0.0388 + 0.616i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.675 + 0.737i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.675 + 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.845585364\)
\(L(\frac12)\) \(\approx\) \(1.845585364\)
\(L(1)\) 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 \( 1 + (0.929 + 0.368i)T \)
good2 \( 1 + (-0.728 - 0.684i)T^{2} \)
3 \( 1 + (-1.69 + 0.933i)T + (0.535 - 0.844i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.425 + 0.904i)T^{2} \)
17 \( 1 + (-0.0627 + 0.998i)T^{2} \)
19 \( 1 + (-0.535 - 0.844i)T^{2} \)
23 \( 1 + (1.11 - 1.35i)T + (-0.187 - 0.982i)T^{2} \)
29 \( 1 + (0.929 - 0.368i)T^{2} \)
31 \( 1 + (0.929 - 0.872i)T + (0.0627 - 0.998i)T^{2} \)
37 \( 1 + (-0.0388 - 0.616i)T + (-0.992 + 0.125i)T^{2} \)
41 \( 1 + (0.187 - 0.982i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (-0.688 + 1.46i)T + (-0.637 - 0.770i)T^{2} \)
53 \( 1 + (-1.84 - 0.233i)T + (0.968 + 0.248i)T^{2} \)
59 \( 1 + (-0.121 - 0.0312i)T + (0.876 + 0.481i)T^{2} \)
61 \( 1 + (0.187 + 0.982i)T^{2} \)
67 \( 1 + (-0.371 + 1.94i)T + (-0.929 - 0.368i)T^{2} \)
71 \( 1 + (0.824 - 1.75i)T + (-0.637 - 0.770i)T^{2} \)
73 \( 1 + (-0.876 + 0.481i)T^{2} \)
79 \( 1 + (-0.535 + 0.844i)T^{2} \)
83 \( 1 + (-0.535 - 0.844i)T^{2} \)
89 \( 1 + (0.824 - 0.211i)T + (0.876 - 0.481i)T^{2} \)
97 \( 1 + (0.0235 + 0.123i)T + (-0.929 + 0.368i)T^{2} \)
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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

−9.291678960941409430635860891825, −8.498172069863132264528238509030, −8.098740681533786213485673383107, −7.42230315773651402544439701681, −6.96823233481356997596464170985, −5.64632090242308363295709046175, −4.03341771062005035857103628121, −3.43335697259508882133500371682, −2.57715608706364383673357699118, −1.54599592651136661385259086604, 2.23674837432607368230048325067, 2.65480237646948388246830923936, 3.79534107220153769052820767152, 4.50532923917120063409263396086, 5.63254448340369198504372639709, 6.97041086790168344317646960087, 7.62287439335752797907717134421, 8.211264862607639919259981540636, 9.115030073621483619365418849495, 10.04384694787498792095521196313

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