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Properties

Label 2-1375-1375.736-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  + (0.0915 − 1.45i)3-s + (−0.637 + 0.770i)4-s + (0.309 − 0.951i)5-s + (−1.11 − 0.141i)9-s + (−0.425 − 0.904i)11-s + (1.06 + 0.998i)12-s + (−1.35 − 0.536i)15-s + (−0.187 − 0.982i)16-s + (0.535 + 0.844i)20-s + (0.110 + 0.0604i)23-s + (−0.809 − 0.587i)25-s + (−0.0343 + 0.179i)27-s + (−1.11 − 1.35i)31-s + (−1.35 + 0.536i)33-s + (0.820 − 0.770i)36-s + (0.303 + 1.58i)37-s + ⋯
L(s)  = 1  + (0.0915 − 1.45i)3-s + (−0.637 + 0.770i)4-s + (0.309 − 0.951i)5-s + (−1.11 − 0.141i)9-s + (−0.425 − 0.904i)11-s + (1.06 + 0.998i)12-s + (−1.35 − 0.536i)15-s + (−0.187 − 0.982i)16-s + (0.535 + 0.844i)20-s + (0.110 + 0.0604i)23-s + (−0.809 − 0.587i)25-s + (−0.0343 + 0.179i)27-s + (−1.11 − 1.35i)31-s + (−1.35 + 0.536i)33-s + (0.820 − 0.770i)36-s + (0.303 + 1.58i)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: $\chi_{1375} (736, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ -0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8628647440\)
\(L(\frac12)\) \(\approx\) \(0.8628647440\)
\(L(1)\) 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 \( 1 + (-0.0915 + 1.45i)T + (-0.992 - 0.125i)T^{2} \)
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 \( 1 + (-0.110 - 0.0604i)T + (0.535 + 0.844i)T^{2} \)
29 \( 1 + (0.425 - 0.904i)T^{2} \)
31 \( 1 + (1.11 + 1.35i)T + (-0.187 + 0.982i)T^{2} \)
37 \( 1 + (-0.303 - 1.58i)T + (-0.929 + 0.368i)T^{2} \)
41 \( 1 + (-0.535 + 0.844i)T^{2} \)
43 \( 1 + (-0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.598 + 0.153i)T + (0.876 - 0.481i)T^{2} \)
53 \( 1 + (-0.791 - 0.313i)T + (0.728 + 0.684i)T^{2} \)
59 \( 1 + (0.273 + 0.256i)T + (0.0627 + 0.998i)T^{2} \)
61 \( 1 + (-0.535 - 0.844i)T^{2} \)
67 \( 1 + (0.996 - 1.57i)T + (-0.425 - 0.904i)T^{2} \)
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 \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \)
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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

−9.140228828229038072895150905253, −8.487566417139118942526077539809, −7.952444041619111616907437847281, −7.27821843060109321448619321091, −6.18608555837043589176177400931, −5.42106993623335899613565815437, −4.38550078927806476764755965121, −3.20505869042121669900469672091, −2.06703270719170011411174762041, −0.72409773050268769876074204520, 2.04133171884518549813673489657, 3.32963589876282090196529420913, 4.18449903149903988219719478696, 5.03334067300504675219538921203, 5.65657359415769605287249018382, 6.71042743963964621441902016207, 7.68100034468931730448711653516, 8.961711727258678395277049965174, 9.421571345430544269937496847147, 10.07446163249866093756527194213

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