Properties

Label 2-1375-1375.571-c0-0-0
Degree $2$
Conductor $1375$
Sign $-0.979 - 0.199i$
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  + (−0.124 − 0.0157i)3-s + (−0.187 − 0.982i)4-s + (−0.809 − 0.587i)5-s + (−0.953 − 0.244i)9-s + (−0.637 + 0.770i)11-s + (0.00788 + 0.125i)12-s + (0.0915 + 0.0859i)15-s + (−0.929 + 0.368i)16-s + (−0.425 + 0.904i)20-s + (−1.06 − 1.67i)23-s + (0.309 + 0.951i)25-s + (0.231 + 0.0917i)27-s + (−0.200 + 1.05i)31-s + (0.0915 − 0.0859i)33-s + (−0.0618 + 0.982i)36-s + (−0.574 + 0.227i)37-s + ⋯
L(s)  = 1  + (−0.124 − 0.0157i)3-s + (−0.187 − 0.982i)4-s + (−0.809 − 0.587i)5-s + (−0.953 − 0.244i)9-s + (−0.637 + 0.770i)11-s + (0.00788 + 0.125i)12-s + (0.0915 + 0.0859i)15-s + (−0.929 + 0.368i)16-s + (−0.425 + 0.904i)20-s + (−1.06 − 1.67i)23-s + (0.309 + 0.951i)25-s + (0.231 + 0.0917i)27-s + (−0.200 + 1.05i)31-s + (0.0915 − 0.0859i)33-s + (−0.0618 + 0.982i)36-s + (−0.574 + 0.227i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.979 - 0.199i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ -0.979 - 0.199i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2427185878\)
\(L(\frac12)\) \(\approx\) \(0.2427185878\)
\(L(1)\) 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 \( 1 + (0.637 - 0.770i)T \)
good2 \( 1 + (0.187 + 0.982i)T^{2} \)
3 \( 1 + (0.124 + 0.0157i)T + (0.968 + 0.248i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.876 - 0.481i)T^{2} \)
17 \( 1 + (0.929 + 0.368i)T^{2} \)
19 \( 1 + (-0.968 + 0.248i)T^{2} \)
23 \( 1 + (1.06 + 1.67i)T + (-0.425 + 0.904i)T^{2} \)
29 \( 1 + (0.637 + 0.770i)T^{2} \)
31 \( 1 + (0.200 - 1.05i)T + (-0.929 - 0.368i)T^{2} \)
37 \( 1 + (0.574 - 0.227i)T + (0.728 - 0.684i)T^{2} \)
41 \( 1 + (0.425 + 0.904i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (1.41 - 0.779i)T + (0.535 - 0.844i)T^{2} \)
53 \( 1 + (0.929 + 0.872i)T + (0.0627 + 0.998i)T^{2} \)
59 \( 1 + (0.116 + 1.85i)T + (-0.992 + 0.125i)T^{2} \)
61 \( 1 + (0.425 - 0.904i)T^{2} \)
67 \( 1 + (0.620 + 1.31i)T + (-0.637 + 0.770i)T^{2} \)
71 \( 1 + (-0.110 + 0.0604i)T + (0.535 - 0.844i)T^{2} \)
73 \( 1 + (0.992 + 0.125i)T^{2} \)
79 \( 1 + (-0.968 - 0.248i)T^{2} \)
83 \( 1 + (-0.968 + 0.248i)T^{2} \)
89 \( 1 + (-0.110 + 1.74i)T + (-0.992 - 0.125i)T^{2} \)
97 \( 1 + (-0.791 + 1.68i)T + (-0.637 - 0.770i)T^{2} \)
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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

−9.333225702245841614023833238040, −8.541984208244744107528480951074, −7.928310452495021302208600630608, −6.76814399227617481877911551569, −6.00592931616038944707113455329, −4.96991587942808371548394650466, −4.56377655606696476937074181484, −3.20434475157921129653507165504, −1.84112987206505368754469864201, −0.19351589102559275432015516175, 2.46256272595576223407420774081, 3.33061068382849150894716030597, 3.97968233166823522946602992928, 5.22569050021758744692946993948, 6.08997963278246694194342808696, 7.21226467002356862190807426968, 7.902961388410951019796766541616, 8.332406479863926065516855417918, 9.235508093108605283931440327203, 10.34038538800422737398710831972

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