Properties

Label 2-55e2-1.1-c1-0-30
Degree $2$
Conductor $3025$
Sign $1$
Analytic cond. $24.1547$
Root an. cond. $4.91474$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 4-s + 2·6-s + 2·7-s + 3·8-s + 9-s + 2·12-s − 13-s − 2·14-s − 16-s + 5·17-s − 18-s + 6·19-s − 4·21-s − 2·23-s − 6·24-s + 26-s + 4·27-s − 2·28-s + 9·29-s − 2·31-s − 5·32-s − 5·34-s − 36-s + 3·37-s − 6·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.577·12-s − 0.277·13-s − 0.534·14-s − 1/4·16-s + 1.21·17-s − 0.235·18-s + 1.37·19-s − 0.872·21-s − 0.417·23-s − 1.22·24-s + 0.196·26-s + 0.769·27-s − 0.377·28-s + 1.67·29-s − 0.359·31-s − 0.883·32-s − 0.857·34-s − 1/6·36-s + 0.493·37-s − 0.973·38-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(24.1547\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7451280270\)
\(L(\frac12)\) \(\approx\) \(0.7451280270\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−8.579461901315169739376678660673, −8.069986483292601761137519952233, −7.37226685783266137740275636717, −6.48190836691386467410771774372, −5.38786861929707559822010222316, −5.14830451796636083715635540878, −4.26835168421269434433264314646, −3.07367978379982844766979708515, −1.49712391443289391355106255202, −0.68832603438404776244692791710, 0.68832603438404776244692791710, 1.49712391443289391355106255202, 3.07367978379982844766979708515, 4.26835168421269434433264314646, 5.14830451796636083715635540878, 5.38786861929707559822010222316, 6.48190836691386467410771774372, 7.37226685783266137740275636717, 8.069986483292601761137519952233, 8.579461901315169739376678660673

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