Properties

Label 2-1375-1375.604-c0-0-0
Degree $2$
Conductor $1375$
Sign $0.910 - 0.414i$
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  + (1.51 − 0.288i)3-s + (−0.876 + 0.481i)4-s + (0.809 + 0.587i)5-s + (1.27 − 0.506i)9-s + (−0.968 + 0.248i)11-s + (−1.18 + 0.982i)12-s + (1.39 + 0.656i)15-s + (0.535 − 0.844i)16-s + (−0.992 − 0.125i)20-s + (1.96 + 0.123i)23-s + (0.309 + 0.951i)25-s + (0.487 − 0.309i)27-s + (−0.110 − 0.0604i)31-s + (−1.39 + 0.656i)33-s + (−0.876 + 1.05i)36-s + (−1.60 − 1.01i)37-s + ⋯
L(s)  = 1  + (1.51 − 0.288i)3-s + (−0.876 + 0.481i)4-s + (0.809 + 0.587i)5-s + (1.27 − 0.506i)9-s + (−0.968 + 0.248i)11-s + (−1.18 + 0.982i)12-s + (1.39 + 0.656i)15-s + (0.535 − 0.844i)16-s + (−0.992 − 0.125i)20-s + (1.96 + 0.123i)23-s + (0.309 + 0.951i)25-s + (0.487 − 0.309i)27-s + (−0.110 − 0.0604i)31-s + (−1.39 + 0.656i)33-s + (−0.876 + 1.05i)36-s + (−1.60 − 1.01i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.910 - 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.613517815\)
\(L(\frac12)\) \(\approx\) \(1.613517815\)
\(L(1)\) 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 \( 1 + (0.968 - 0.248i)T \)
good2 \( 1 + (0.876 - 0.481i)T^{2} \)
3 \( 1 + (-1.51 + 0.288i)T + (0.929 - 0.368i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.728 - 0.684i)T^{2} \)
17 \( 1 + (0.535 + 0.844i)T^{2} \)
19 \( 1 + (0.929 + 0.368i)T^{2} \)
23 \( 1 + (-1.96 - 0.123i)T + (0.992 + 0.125i)T^{2} \)
29 \( 1 + (-0.968 - 0.248i)T^{2} \)
31 \( 1 + (0.110 + 0.0604i)T + (0.535 + 0.844i)T^{2} \)
37 \( 1 + (1.60 + 1.01i)T + (0.425 + 0.904i)T^{2} \)
41 \( 1 + (0.992 - 0.125i)T^{2} \)
43 \( 1 + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (0.804 - 0.856i)T + (-0.0627 - 0.998i)T^{2} \)
53 \( 1 + (0.450 + 0.211i)T + (0.637 + 0.770i)T^{2} \)
59 \( 1 + (0.683 + 0.825i)T + (-0.187 + 0.982i)T^{2} \)
61 \( 1 + (0.992 + 0.125i)T^{2} \)
67 \( 1 + (-0.226 - 1.79i)T + (-0.968 + 0.248i)T^{2} \)
71 \( 1 + (0.929 + 0.872i)T + (0.0627 + 0.998i)T^{2} \)
73 \( 1 + (-0.187 - 0.982i)T^{2} \)
79 \( 1 + (0.929 - 0.368i)T^{2} \)
83 \( 1 + (-0.929 - 0.368i)T^{2} \)
89 \( 1 + (-0.929 + 1.12i)T + (-0.187 - 0.982i)T^{2} \)
97 \( 1 + (-0.211 + 1.67i)T + (-0.968 - 0.248i)T^{2} \)
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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

−9.595885198618763562144388017888, −8.964051464972904746197295806895, −8.385270987636906087212472048471, −7.46419086784300371883149052735, −6.99332605023990607358652512998, −5.55043717825936169971523184052, −4.69747288081645042939673406344, −3.35808728264650384511341274581, −2.94220625188514426837170626179, −1.83028878945929911818838507639, 1.43138466614939977835735629254, 2.66119136003002778838579881888, 3.54777835299527673804097260405, 4.81744805521547124034319114347, 5.17334060566924269060645682994, 6.39147583236679313518246890217, 7.64185075771424861171486025197, 8.483272249725813001617830432273, 8.916586080593156424804024609305, 9.499267480964940934150816265622

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