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Properties

Label 2-3875-3875.929-c0-0-1
Degree $2$
Conductor $3875$
Sign $-0.979 - 0.199i$
Analytic cond. $1.93387$
Root an. cond. $1.39063$
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  + (−1.30 − 1.07i)2-s + (0.346 + 1.81i)4-s + (−0.0627 − 0.998i)5-s + (−0.147 + 0.202i)7-s + (0.692 − 1.26i)8-s + (−0.968 − 0.248i)9-s + (−0.992 + 1.36i)10-s + (0.409 − 0.105i)14-s + (−0.536 + 0.212i)16-s + (0.992 + 1.36i)18-s + (1.44 − 0.182i)19-s + (1.79 − 0.460i)20-s + (−0.992 + 0.125i)25-s + (−0.419 − 0.197i)28-s + (0.187 − 0.982i)31-s + (−0.441 − 0.143i)32-s + ⋯
L(s)  = 1  + (−1.30 − 1.07i)2-s + (0.346 + 1.81i)4-s + (−0.0627 − 0.998i)5-s + (−0.147 + 0.202i)7-s + (0.692 − 1.26i)8-s + (−0.968 − 0.248i)9-s + (−0.992 + 1.36i)10-s + (0.409 − 0.105i)14-s + (−0.536 + 0.212i)16-s + (0.992 + 1.36i)18-s + (1.44 − 0.182i)19-s + (1.79 − 0.460i)20-s + (−0.992 + 0.125i)25-s + (−0.419 − 0.197i)28-s + (0.187 − 0.982i)31-s + (−0.441 − 0.143i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3875\)    =    \(5^{3} \cdot 31\)
Sign: $-0.979 - 0.199i$
Analytic conductor: \(1.93387\)
Root analytic conductor: \(1.39063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3875} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3875,\ (\ :0),\ -0.979 - 0.199i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3963873470\)
\(L(\frac12)\) \(\approx\) \(0.3963873470\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.0627 + 0.998i)T \)
31 \( 1 + (-0.187 + 0.982i)T \)
good2 \( 1 + (1.30 + 1.07i)T + (0.187 + 0.982i)T^{2} \)
3 \( 1 + (0.968 + 0.248i)T^{2} \)
7 \( 1 + (0.147 - 0.202i)T + (-0.309 - 0.951i)T^{2} \)
11 \( 1 + (0.187 + 0.982i)T^{2} \)
13 \( 1 + (0.876 + 0.481i)T^{2} \)
17 \( 1 + (-0.929 - 0.368i)T^{2} \)
19 \( 1 + (-1.44 + 0.182i)T + (0.968 - 0.248i)T^{2} \)
23 \( 1 + (-0.425 + 0.904i)T^{2} \)
29 \( 1 + (0.637 + 0.770i)T^{2} \)
37 \( 1 + (0.728 - 0.684i)T^{2} \)
41 \( 1 + (-0.939 + 1.47i)T + (-0.425 - 0.904i)T^{2} \)
43 \( 1 + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (0.354 + 0.645i)T + (-0.535 + 0.844i)T^{2} \)
53 \( 1 + (0.0627 + 0.998i)T^{2} \)
59 \( 1 + (0.124 + 1.98i)T + (-0.992 + 0.125i)T^{2} \)
61 \( 1 + (0.425 - 0.904i)T^{2} \)
67 \( 1 + (1.63 - 0.770i)T + (0.637 - 0.770i)T^{2} \)
71 \( 1 + (1.75 - 0.963i)T + (0.535 - 0.844i)T^{2} \)
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 \( 1 + (0.992 + 0.125i)T^{2} \)
97 \( 1 + (1.80 + 0.849i)T + (0.637 + 0.770i)T^{2} \)
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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

−8.541102167572556149511631319828, −7.889017347294091251236416931225, −7.23134428898867307110907593485, −5.90635898946471036975128975984, −5.35412496023108682728750749874, −4.18302488325550161716592078034, −3.25158215879844409128448343548, −2.49934023452051023995205460312, −1.43194728486959188589985001187, −0.38632102073143616446260534567, 1.28650241779860241112296486600, 2.70578917227809373347360803614, 3.43229711483752876482976459172, 4.84123412331787178272848135436, 5.84860618160576325417998895624, 6.18352912473284591149901244735, 7.12915187520680691113061434212, 7.54801096812272079801382635133, 8.175358769269473972353945781788, 8.966962398549946702853195914888

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