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Properties

Label 2-55e2-25.23-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.992 - 0.125i$
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 + 0.142i)3-s + (0.951 + 0.309i)4-s − i·5-s + (−0.530 + 0.729i)9-s + (−0.309 + 0.0489i)12-s + (0.142 + 0.278i)15-s + (0.809 + 0.587i)16-s + (0.309 − 0.951i)20-s + (1.76 + 0.278i)23-s − 25-s + (0.0930 − 0.587i)27-s + (0.363 + 1.11i)31-s + (−0.729 + 0.530i)36-s + (1.39 − 0.221i)37-s + (0.729 + 0.530i)45-s + ⋯
L(s)  = 1  + (−0.278 + 0.142i)3-s + (0.951 + 0.309i)4-s − i·5-s + (−0.530 + 0.729i)9-s + (−0.309 + 0.0489i)12-s + (0.142 + 0.278i)15-s + (0.809 + 0.587i)16-s + (0.309 − 0.951i)20-s + (1.76 + 0.278i)23-s − 25-s + (0.0930 − 0.587i)27-s + (0.363 + 1.11i)31-s + (−0.729 + 0.530i)36-s + (1.39 − 0.221i)37-s + (0.729 + 0.530i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.992 - 0.125i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (848, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.426775448\)
\(L(\frac12)\) \(\approx\) \(1.426775448\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
11 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T^{2} \)
3 \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.39 + 0.221i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.642 + 1.26i)T + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.412i)T + (0.587 + 0.809i)T^{2} \)
71 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
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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

−8.683648683371853369694088575914, −8.283425615242074139406316821820, −7.40548454048762069068996129022, −6.72988649097654986278823594801, −5.71822317352587418222198568359, −5.20505421829981140995199430561, −4.34062197131309767804191279687, −3.21517113233525612745362183440, −2.33494449684898013815295087693, −1.20782085047080440385464070045, 1.10339402427894535005551570400, 2.54589570566218207007203370501, 2.96877562441527023071446164379, 4.05131709510177680143412021467, 5.33104587774099527298733548018, 6.07538218975742500158783553660, 6.57360650017422792001627908528, 7.21750502634831771386215362765, 7.898341737628156263913217460608, 8.982991047575409140341699149058

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