Properties

Label 2-1305-1.1-c1-0-29
Degree $2$
Conductor $1305$
Sign $-1$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.772·2-s − 1.40·4-s − 5-s + 2.17·7-s + 2.62·8-s + 0.772·10-s − 3·11-s − 0.629·13-s − 1.68·14-s + 0.772·16-s − 4.17·17-s + 4.80·19-s + 1.40·20-s + 2.31·22-s − 2.08·23-s + 25-s + 0.486·26-s − 3.05·28-s + 29-s − 5.85·32-s + 3.22·34-s − 2.17·35-s + 6.08·37-s − 3.71·38-s − 2.62·40-s − 0.824·41-s − 8.72·43-s + ⋯
L(s)  = 1  − 0.546·2-s − 0.701·4-s − 0.447·5-s + 0.822·7-s + 0.929·8-s + 0.244·10-s − 0.904·11-s − 0.174·13-s − 0.449·14-s + 0.193·16-s − 1.01·17-s + 1.10·19-s + 0.313·20-s + 0.494·22-s − 0.434·23-s + 0.200·25-s + 0.0954·26-s − 0.576·28-s + 0.185·29-s − 1.03·32-s + 0.553·34-s − 0.367·35-s + 1.00·37-s − 0.602·38-s − 0.415·40-s − 0.128·41-s − 1.32·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $-1$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 + T \)
29 \( 1 - T \)
good2 \( 1 + 0.772T + 2T^{2} \)
7 \( 1 - 2.17T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 0.629T + 13T^{2} \)
17 \( 1 + 4.17T + 17T^{2} \)
19 \( 1 - 4.80T + 19T^{2} \)
23 \( 1 + 2.08T + 23T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6.08T + 37T^{2} \)
41 \( 1 + 0.824T + 41T^{2} \)
43 \( 1 + 8.72T + 43T^{2} \)
47 \( 1 - 8.98T + 47T^{2} \)
53 \( 1 + 6.88T + 53T^{2} \)
59 \( 1 + 6.45T + 59T^{2} \)
61 \( 1 + 2.80T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 + 2.63T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 - 7.62T + 83T^{2} \)
89 \( 1 + 17.8T + 89T^{2} \)
97 \( 1 - 0.538T + 97T^{2} \)
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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

−9.173119287036945031206222681773, −8.366663494210071413922595888563, −7.82574119170216455372999927467, −7.14806405725519381589882957738, −5.75176145925009091942338120077, −4.83054435844237780996654853200, −4.28320081259642496990869590765, −2.93092831436610620795706668970, −1.50569963813351064387122547282, 0, 1.50569963813351064387122547282, 2.93092831436610620795706668970, 4.28320081259642496990869590765, 4.83054435844237780996654853200, 5.75176145925009091942338120077, 7.14806405725519381589882957738, 7.82574119170216455372999927467, 8.366663494210071413922595888563, 9.173119287036945031206222681773

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