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Properties

Label 2-1375-1375.296-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  + (−1.35 − 1.27i)3-s + (−0.425 + 0.904i)4-s + (−0.809 − 0.587i)5-s + (0.154 + 2.45i)9-s + (0.535 + 0.844i)11-s + (1.72 − 0.684i)12-s + (0.348 + 1.82i)15-s + (−0.637 − 0.770i)16-s + (0.876 − 0.481i)20-s + (1.41 − 0.362i)23-s + (0.309 + 0.951i)25-s + (1.72 − 2.08i)27-s + (−0.824 − 1.75i)31-s + (0.348 − 1.82i)33-s + (−2.28 − 0.904i)36-s + (−0.393 − 0.476i)37-s + ⋯
L(s)  = 1  + (−1.35 − 1.27i)3-s + (−0.425 + 0.904i)4-s + (−0.809 − 0.587i)5-s + (0.154 + 2.45i)9-s + (0.535 + 0.844i)11-s + (1.72 − 0.684i)12-s + (0.348 + 1.82i)15-s + (−0.637 − 0.770i)16-s + (0.876 − 0.481i)20-s + (1.41 − 0.362i)23-s + (0.309 + 0.951i)25-s + (1.72 − 2.08i)27-s + (−0.824 − 1.75i)31-s + (0.348 − 1.82i)33-s + (−2.28 − 0.904i)36-s + (−0.393 − 0.476i)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: $\chi_{1375} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.675 + 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4991377248\)
\(L(\frac12)\) \(\approx\) \(0.4991377248\)
\(L(1)\) 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 \( 1 + (-0.535 - 0.844i)T \)
good2 \( 1 + (0.425 - 0.904i)T^{2} \)
3 \( 1 + (1.35 + 1.27i)T + (0.0627 + 0.998i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.992 - 0.125i)T^{2} \)
17 \( 1 + (0.637 - 0.770i)T^{2} \)
19 \( 1 + (-0.0627 + 0.998i)T^{2} \)
23 \( 1 + (-1.41 + 0.362i)T + (0.876 - 0.481i)T^{2} \)
29 \( 1 + (-0.535 + 0.844i)T^{2} \)
31 \( 1 + (0.824 + 1.75i)T + (-0.637 + 0.770i)T^{2} \)
37 \( 1 + (0.393 + 0.476i)T + (-0.187 + 0.982i)T^{2} \)
41 \( 1 + (-0.876 - 0.481i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (-1.60 - 0.202i)T + (0.968 + 0.248i)T^{2} \)
53 \( 1 + (0.200 + 1.05i)T + (-0.929 + 0.368i)T^{2} \)
59 \( 1 + (-1.18 + 0.469i)T + (0.728 - 0.684i)T^{2} \)
61 \( 1 + (-0.876 + 0.481i)T^{2} \)
67 \( 1 + (0.328 + 0.180i)T + (0.535 + 0.844i)T^{2} \)
71 \( 1 + (-1.84 - 0.233i)T + (0.968 + 0.248i)T^{2} \)
73 \( 1 + (-0.728 - 0.684i)T^{2} \)
79 \( 1 + (-0.0627 - 0.998i)T^{2} \)
83 \( 1 + (-0.0627 + 0.998i)T^{2} \)
89 \( 1 + (-1.84 - 0.730i)T + (0.728 + 0.684i)T^{2} \)
97 \( 1 + (1.11 - 0.614i)T + (0.535 - 0.844i)T^{2} \)
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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

−9.514120290877064531024790608519, −8.677546343256045013837119817101, −7.73475965277949572447680847011, −7.32190759828275867835873440651, −6.62999567677407231286747515688, −5.44357496926003604562616343887, −4.71447791752343937233357213628, −3.85622807786930740516311697576, −2.23338645253155605948969509622, −0.76271225145232390612741148376, 0.883528886354449571503620346403, 3.31889152758805563504341411386, 4.03426142837890081669475885831, 4.96389731776238581209429691199, 5.54258733906692808888851984363, 6.43681930453425564864361273037, 7.08008443955647873885199237510, 8.729225341716443461751779895590, 9.150812705023181986445896470889, 10.22603598125253228139715356111

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