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Properties

Label 2-1925-385.104-c0-0-2
Degree $2$
Conductor $1925$
Sign $0.865 + 0.500i$
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  + (−1.53 + 0.5i)2-s + (1.30 − 0.951i)4-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (−1.30 − 0.951i)14-s + (0.951 + 1.30i)18-s + 1.61i·22-s − 1.61i·23-s + (1.53 + 0.5i)28-s + (0.5 − 0.363i)29-s − 0.999i·32-s + (−1.30 − 0.951i)36-s + (−0.951 − 1.30i)37-s + ⋯
L(s)  = 1  + (−1.53 + 0.5i)2-s + (1.30 − 0.951i)4-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (−1.30 − 0.951i)14-s + (0.951 + 1.30i)18-s + 1.61i·22-s − 1.61i·23-s + (1.53 + 0.5i)28-s + (0.5 − 0.363i)29-s − 0.999i·32-s + (−1.30 − 0.951i)36-s + (−0.951 − 1.30i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.865 + 0.500i$
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: \((2,\ 1925,\ (\ :0),\ 0.865 + 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5241039537\)
\(L(\frac12)\) \(\approx\) \(0.5241039537\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
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 \( 1 + 1.61iT - T^{2} \)
29 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 0.618iT - T^{2} \)
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 \( 1 + 0.618iT - T^{2} \)
71 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.048538205218657649854187838988, −8.617322617389032692477595023901, −8.110836786056610078045127387366, −7.09441523359880415589109989726, −6.26160554698847317442157045202, −5.81982025886128521682023499532, −4.50801862896089886944466612267, −3.20338093931761634288352065373, −2.02091613764793395320639794725, −0.69363410392161810517184693472, 1.35709836620588678657737475521, 2.06397927176916924294689299075, 3.31106833559313285185184828603, 4.56306291858931450229657120029, 5.34202645984028283795235510287, 6.83969495958660580825178642060, 7.38771908676494688128433138584, 8.028180498744098610068568176447, 8.663368355461988866589486154724, 9.571202381503659273571737014166

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