Properties

Label 2-118580-1.1-c1-0-25
Degree $2$
Conductor $118580$
Sign $-1$
Analytic cond. $946.866$
Root an. cond. $30.7711$
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  − 3·3-s + 5-s + 6·9-s − 3·13-s − 3·15-s − 17-s + 6·19-s + 6·23-s + 25-s − 9·27-s + 9·29-s + 4·31-s + 2·37-s + 9·39-s − 4·41-s − 10·43-s + 6·45-s + 47-s + 3·51-s + 4·53-s − 18·57-s + 8·59-s − 8·61-s − 3·65-s + 12·67-s − 18·69-s + 8·71-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 2·9-s − 0.832·13-s − 0.774·15-s − 0.242·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s + 0.718·31-s + 0.328·37-s + 1.44·39-s − 0.624·41-s − 1.52·43-s + 0.894·45-s + 0.145·47-s + 0.420·51-s + 0.549·53-s − 2.38·57-s + 1.04·59-s − 1.02·61-s − 0.372·65-s + 1.46·67-s − 2.16·69-s + 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 118580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118580 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(118580\)    =    \(2^{2} \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(946.866\)
Root analytic conductor: \(30.7711\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 118580,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.71082697874367, −13.27602074623255, −12.63073121593453, −12.37486287881056, −11.73883520536276, −11.49349279542079, −11.07590367631564, −10.24857796981473, −10.12944115436754, −9.724105880309859, −9.013939936760954, −8.422713677922032, −7.736002954187432, −7.065177411932629, −6.781490601146915, −6.371025204799398, −5.642674205507643, −5.253812290451795, −4.812332434515617, −4.498634265750719, −3.515730883095756, −2.880761211723664, −2.188206954989884, −1.190809856243946, −0.9236783324041348, 0, 0.9236783324041348, 1.190809856243946, 2.188206954989884, 2.880761211723664, 3.515730883095756, 4.498634265750719, 4.812332434515617, 5.253812290451795, 5.642674205507643, 6.371025204799398, 6.781490601146915, 7.065177411932629, 7.736002954187432, 8.422713677922032, 9.013939936760954, 9.724105880309859, 10.12944115436754, 10.24857796981473, 11.07590367631564, 11.49349279542079, 11.73883520536276, 12.37486287881056, 12.63073121593453, 13.27602074623255, 13.71082697874367

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