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Properties

Label 2-55e2-275.147-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.689 - 0.724i$
Analytic cond. $1.50967$
Root an. cond. $1.22868$
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  + (−0.278 + 1.76i)3-s + (−0.951 − 0.309i)4-s + (−0.587 − 0.809i)5-s + (−2.06 − 0.672i)9-s + (0.809 − 1.58i)12-s + (1.58 − 0.809i)15-s + (0.809 + 0.587i)16-s + (0.309 + 0.951i)20-s + (−0.896 − 1.76i)23-s + (−0.309 + 0.951i)25-s + (0.951 − 1.86i)27-s + 1.90·31-s + (1.76 + 1.27i)36-s + (1.39 + 0.221i)37-s + (0.672 + 2.06i)45-s + ⋯
L(s)  = 1  + (−0.278 + 1.76i)3-s + (−0.951 − 0.309i)4-s + (−0.587 − 0.809i)5-s + (−2.06 − 0.672i)9-s + (0.809 − 1.58i)12-s + (1.58 − 0.809i)15-s + (0.809 + 0.587i)16-s + (0.309 + 0.951i)20-s + (−0.896 − 1.76i)23-s + (−0.309 + 0.951i)25-s + (0.951 − 1.86i)27-s + 1.90·31-s + (1.76 + 1.27i)36-s + (1.39 + 0.221i)37-s + (0.672 + 2.06i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.689 - 0.724i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (2622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.689 - 0.724i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6716414328\)
\(L(\frac12)\) \(\approx\) \(0.6716414328\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T^{2} \)
3 \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + (-0.587 - 0.809i)T^{2} \)
13 \( 1 + (-0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 - 1.90T + T^{2} \)
37 \( 1 + (-1.39 - 0.221i)T + (0.951 + 0.309i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \)
71 \( 1 - 0.618T + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T^{2} \)
89 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.412 - 0.809i)T + (-0.587 - 0.809i)T^{2} \)
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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

−9.061768350576064149794836831540, −8.460198618818816638329139623676, −8.009754883376753281728055276075, −6.39850684840307567240448955556, −5.63498575290636378967427503471, −4.86577974291822908840111613049, −4.33090099171840656004248599203, −3.98622486026833049790290896949, −2.80567334368150643356072340119, −0.72697308775844914042274682404, 0.76614987799105969735255107414, 2.10502158514400259403998925656, 3.10122238820921117308334786008, 3.94966696114166990744072921515, 5.10219096055733221869412465513, 5.98758326606142307682479768288, 6.65184689411257251236088921411, 7.40607796830105413789817693443, 8.057837669511094232951724900591, 8.278622510354865774291939897923

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