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Properties

Label 2-55e2-275.38-c0-0-0
Degree $2$
Conductor $3025$
Sign $-0.668 + 0.743i$
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  + (−1.76 + 0.896i)3-s + (0.587 − 0.809i)4-s + (−0.951 − 0.309i)5-s + (1.70 − 2.34i)9-s + (−0.309 + 1.95i)12-s + (1.95 − 0.309i)15-s + (−0.309 − 0.951i)16-s + (−0.809 + 0.587i)20-s + (−0.142 − 0.896i)23-s + (0.809 + 0.587i)25-s + (−0.587 + 3.71i)27-s − 1.17·31-s + (−0.896 − 2.76i)36-s + (0.642 + 1.26i)37-s + (−2.34 + 1.70i)45-s + ⋯
L(s)  = 1  + (−1.76 + 0.896i)3-s + (0.587 − 0.809i)4-s + (−0.951 − 0.309i)5-s + (1.70 − 2.34i)9-s + (−0.309 + 1.95i)12-s + (1.95 − 0.309i)15-s + (−0.309 − 0.951i)16-s + (−0.809 + 0.587i)20-s + (−0.142 − 0.896i)23-s + (0.809 + 0.587i)25-s + (−0.587 + 3.71i)27-s − 1.17·31-s + (−0.896 − 2.76i)36-s + (0.642 + 1.26i)37-s + (−2.34 + 1.70i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.668 + 0.743i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1963, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ -0.668 + 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3072734385\)
\(L(\frac12)\) \(\approx\) \(0.3072734385\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T^{2} \)
3 \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \)
7 \( 1 + (-0.951 - 0.309i)T^{2} \)
13 \( 1 + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + 1.17T + T^{2} \)
37 \( 1 + (-0.642 - 1.26i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.642 + 1.26i)T + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \)
71 \( 1 + 1.61T + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.0489 - 0.309i)T + (-0.951 - 0.309i)T^{2} \)
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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

−8.878536486679409554971652817523, −7.63735097661256490142887798787, −6.83912157112332366517385819078, −6.26111182773067262562828093667, −5.50821864281541697263859581961, −4.82359681263023660097617985151, −4.26778561696593720896333518597, −3.23202721853379568557912414901, −1.44253445018441261701359419537, −0.25774165824788508797429561501, 1.39883454688163023224410367790, 2.54927966826502893041674817878, 3.79321238149767573855030848731, 4.55429348120699320342407484457, 5.60427961016837781646466088185, 6.24191372362962312400557554478, 7.03848314338901835657815378345, 7.53038497800988193487881757929, 7.87318221405585733804441047920, 9.038973818274955811742649699835

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