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Properties

Label 2-1925-385.104-c0-0-1
Degree $2$
Conductor $1925$
Sign $0.393 - 0.919i$
Analytic cond. $0.960700$
Root an. cond. $0.980153$
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  + (−0.198 + 0.0646i)2-s + (−0.773 + 0.562i)4-s + (0.587 + 0.809i)7-s + (0.240 − 0.330i)8-s + (−0.309 − 0.951i)9-s + (0.669 + 0.743i)11-s + (−0.169 − 0.122i)14-s + (0.269 − 0.828i)16-s + (0.122 + 0.169i)18-s + (−0.181 − 0.104i)22-s + 1.82i·23-s + (−0.909 − 0.295i)28-s + (1.08 − 0.786i)29-s + 0.591i·32-s + (0.773 + 0.562i)36-s + (−0.122 − 0.169i)37-s + ⋯
L(s)  = 1  + (−0.198 + 0.0646i)2-s + (−0.773 + 0.562i)4-s + (0.587 + 0.809i)7-s + (0.240 − 0.330i)8-s + (−0.309 − 0.951i)9-s + (0.669 + 0.743i)11-s + (−0.169 − 0.122i)14-s + (0.269 − 0.828i)16-s + (0.122 + 0.169i)18-s + (−0.181 − 0.104i)22-s + 1.82i·23-s + (−0.909 − 0.295i)28-s + (1.08 − 0.786i)29-s + 0.591i·32-s + (0.773 + 0.562i)36-s + (−0.122 − 0.169i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.393 - 0.919i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.393 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9075112557\)
\(L(\frac12)\) \(\approx\) \(0.9075112557\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
good2 \( 1 + (0.198 - 0.0646i)T + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - 1.82iT - T^{2} \)
29 \( 1 + (-1.08 + 0.786i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.122 + 0.169i)T + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.33iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - 1.95iT - T^{2} \)
71 \( 1 + (-0.413 + 1.27i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.604 - 1.86i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−9.511671182745643516057868967493, −8.786519874402717813637064811366, −8.109754941579543712176201154593, −7.34049975747880643163926467882, −6.38272744104710490835970429045, −5.48410664224707796527943229079, −4.58525953666361917481012331363, −3.79185533193946554908061555135, −2.81403870984568147807046018865, −1.36913569589645218420254969463, 0.843109429743257313149247541705, 2.05905439374893448847838865473, 3.48555012374205895394155921173, 4.55774151968428799730892051208, 4.97448579353557705794844870861, 6.03099721411714879117063215191, 6.86471333582252229981279487663, 7.964217933155051401119316494945, 8.497868558485941895156387965882, 9.078800696887827032044028001950

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