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Properties

Label 2-129360-1.1-c1-0-108
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s + 11-s + 7·13-s − 15-s + 2·17-s − 2·23-s + 25-s + 27-s + 2·29-s + 3·31-s + 33-s + 12·37-s + 7·39-s + 6·41-s − 43-s − 45-s + 10·47-s + 2·51-s − 55-s − 7·59-s − 7·65-s + 4·67-s − 2·69-s − 9·71-s + 9·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.94·13-s − 0.258·15-s + 0.485·17-s − 0.417·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.538·31-s + 0.174·33-s + 1.97·37-s + 1.12·39-s + 0.937·41-s − 0.152·43-s − 0.149·45-s + 1.45·47-s + 0.280·51-s − 0.134·55-s − 0.911·59-s − 0.868·65-s + 0.488·67-s − 0.240·69-s − 1.06·71-s + 1.05·73-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: \(0\)
Selberg data: \((2,\ 129360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.577701119\)
\(L(\frac12)\) \(\approx\) \(4.577701119\)
\(L(\frac{3}{2})\) 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 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.52475319668425, −13.08429638607244, −12.57891490280706, −12.04610896318937, −11.57273318558088, −11.00422401842054, −10.72708519616233, −10.08441084312749, −9.476880296561730, −9.101975922533578, −8.526397400110023, −8.114041761782120, −7.756653824334342, −7.129962627514689, −6.490606726254446, −5.995673882679352, −5.671739756149029, −4.674544143665039, −4.209263851915511, −3.813188215481167, −3.200727698620536, −2.689710581249765, −1.903759050294294, −1.108421836407083, −0.7356496715920607, 0.7356496715920607, 1.108421836407083, 1.903759050294294, 2.689710581249765, 3.200727698620536, 3.813188215481167, 4.209263851915511, 4.674544143665039, 5.671739756149029, 5.995673882679352, 6.490606726254446, 7.129962627514689, 7.756653824334342, 8.114041761782120, 8.526397400110023, 9.101975922533578, 9.476880296561730, 10.08441084312749, 10.72708519616233, 11.00422401842054, 11.57273318558088, 12.04610896318937, 12.57891490280706, 13.08429638607244, 13.52475319668425

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