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Properties

Label 2-129360-1.1-c1-0-176
Degree $2$
Conductor $129360$
Sign $-1$
Analytic cond. $1032.94$
Root an. cond. $32.1394$
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-s + 5-s + 9-s − 11-s + 2·13-s + 15-s − 8·17-s + 6·23-s + 25-s + 27-s − 4·29-s + 4·31-s − 33-s + 4·37-s + 2·39-s − 6·41-s − 4·43-s + 45-s + 8·47-s − 8·51-s − 4·53-s − 55-s + 10·59-s − 2·61-s + 2·65-s − 4·67-s + 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s − 1.94·17-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.718·31-s − 0.174·33-s + 0.657·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s − 1.12·51-s − 0.549·53-s − 0.134·55-s + 1.30·59-s − 0.256·61-s + 0.248·65-s − 0.488·67-s + 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129360\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(1032.94\)
Root analytic conductor: \(32.1394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 129360,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.56336232529491, −13.24397733622850, −13.10525298025993, −12.37906966408789, −11.80463001180042, −11.12614709170500, −10.89757578007144, −10.43502307353832, −9.581710339529783, −9.487395279388402, −8.699970139485248, −8.563900678673111, −7.943761833761110, −7.247981433896901, −6.733860662748454, −6.476899149391167, −5.694428945353691, −5.156643139416362, −4.590052028025178, −4.060734542251743, −3.454967943415208, −2.710704177267384, −2.364381719404367, −1.663210754549352, −0.9685568964488906, 0, 0.9685568964488906, 1.663210754549352, 2.364381719404367, 2.710704177267384, 3.454967943415208, 4.060734542251743, 4.590052028025178, 5.156643139416362, 5.694428945353691, 6.476899149391167, 6.733860662748454, 7.247981433896901, 7.943761833761110, 8.563900678673111, 8.699970139485248, 9.487395279388402, 9.581710339529783, 10.43502307353832, 10.89757578007144, 11.12614709170500, 11.80463001180042, 12.37906966408789, 13.10525298025993, 13.24397733622850, 13.56336232529491

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