Properties

Label 2-129360-1.1-c1-0-126
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 − 8·23-s + 25-s − 27-s + 2·29-s − 33-s − 6·37-s − 2·39-s − 6·41-s + 8·43-s + 45-s − 4·47-s + 2·51-s − 6·53-s + 55-s + 4·59-s + 6·61-s + 2·65-s − 8·67-s + 8·69-s − 12·71-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 − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.174·33-s − 0.986·37-s − 0.320·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s − 0.583·47-s + 0.280·51-s − 0.824·53-s + 0.134·55-s + 0.520·59-s + 0.768·61-s + 0.248·65-s − 0.977·67-s + 0.963·69-s − 1.42·71-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 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.68752247384624, −13.26915127061580, −12.77652232172843, −12.15870717002738, −11.89626284031609, −11.31215413972863, −10.87358884486376, −10.24994521844517, −10.04654942092012, −9.408943056727228, −8.821728350703747, −8.450917355195085, −7.811409857428064, −7.253741159582841, −6.628875257687492, −6.257816908268527, −5.806980081882975, −5.273951012384032, −4.619879323105517, −4.140539362414537, −3.551161181170911, −2.890843240701002, −1.994187264323898, −1.691038348487026, −0.8094835699411009, 0, 0.8094835699411009, 1.691038348487026, 1.994187264323898, 2.890843240701002, 3.551161181170911, 4.140539362414537, 4.619879323105517, 5.273951012384032, 5.806980081882975, 6.257816908268527, 6.628875257687492, 7.253741159582841, 7.811409857428064, 8.450917355195085, 8.821728350703747, 9.408943056727228, 10.04654942092012, 10.24994521844517, 10.87358884486376, 11.31215413972863, 11.89626284031609, 12.15870717002738, 12.77652232172843, 13.26915127061580, 13.68752247384624

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