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Properties

Label 2-129360-1.1-c1-0-128
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 − 2·17-s − 4·19-s + 25-s + 27-s − 2·29-s − 8·31-s − 33-s − 2·37-s + 2·39-s + 6·41-s − 4·43-s − 45-s + 8·47-s − 2·51-s − 10·53-s + 55-s − 4·57-s − 4·59-s + 2·61-s − 2·65-s + 12·67-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 − 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.174·33-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s − 1.37·53-s + 0.134·55-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.248·65-s + 1.46·67-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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.72955999471194, −13.22348524724888, −12.71773747274896, −12.53018778924850, −11.81430596845609, −11.10630149713632, −10.96349134114113, −10.44293801879371, −9.790488574891583, −9.214886954473407, −8.849938955267278, −8.412107623603053, −7.765453768284041, −7.523521905000035, −6.790992347056442, −6.383587196068805, −5.740518943319274, −5.121876052080898, −4.533779169802623, −3.934094223328044, −3.572092156475960, −2.912428210040645, −2.172694586146284, −1.774798176388649, −0.8146236226830016, 0, 0.8146236226830016, 1.774798176388649, 2.172694586146284, 2.912428210040645, 3.572092156475960, 3.934094223328044, 4.533779169802623, 5.121876052080898, 5.740518943319274, 6.383587196068805, 6.790992347056442, 7.523521905000035, 7.765453768284041, 8.412107623603053, 8.849938955267278, 9.214886954473407, 9.790488574891583, 10.44293801879371, 10.96349134114113, 11.10630149713632, 11.81430596845609, 12.53018778924850, 12.71773747274896, 13.22348524724888, 13.72955999471194

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