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Properties

Label 2-58835-1.1-c1-0-10
Degree $2$
Conductor $58835$
Sign $-1$
Analytic cond. $469.799$
Root an. cond. $21.6748$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s + 5-s + 4·6-s + 7-s + 9-s − 2·10-s + 4·11-s − 4·12-s − 2·14-s − 2·15-s − 4·16-s + 4·17-s − 2·18-s − 5·19-s + 2·20-s − 2·21-s − 8·22-s − 4·23-s + 25-s + 4·27-s + 2·28-s + 6·29-s + 4·30-s − 2·31-s + 8·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s + 0.447·5-s + 1.63·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 1.15·12-s − 0.534·14-s − 0.516·15-s − 16-s + 0.970·17-s − 0.471·18-s − 1.14·19-s + 0.447·20-s − 0.436·21-s − 1.70·22-s − 0.834·23-s + 1/5·25-s + 0.769·27-s + 0.377·28-s + 1.11·29-s + 0.730·30-s − 0.359·31-s + 1.41·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58835 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58835\)    =    \(5 \cdot 7 \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(469.799\)
Root analytic conductor: \(21.6748\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 58835,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 - T \)
41 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.55993618947710, −14.05255492257114, −13.78984683540811, −12.78291744439541, −12.37022147962112, −11.86402163026503, −11.35867406423702, −11.00865154033681, −10.36077984262608, −10.01547202468264, −9.612221240291799, −8.856885581357009, −8.419678805288715, −8.086725906028650, −7.206897674791878, −6.717632852378269, −6.282241072996566, −5.829491778301178, −4.932698562439343, −4.655224301038980, −3.776988368468424, −2.963979633628120, −1.822649237280184, −1.603664910982963, −0.7692428838787727, 0, 0.7692428838787727, 1.603664910982963, 1.822649237280184, 2.963979633628120, 3.776988368468424, 4.655224301038980, 4.932698562439343, 5.829491778301178, 6.282241072996566, 6.717632852378269, 7.206897674791878, 8.086725906028650, 8.419678805288715, 8.856885581357009, 9.612221240291799, 10.01547202468264, 10.36077984262608, 11.00865154033681, 11.35867406423702, 11.86402163026503, 12.37022147962112, 12.78291744439541, 13.78984683540811, 14.05255492257114, 14.55993618947710

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