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Properties

Label 2-129360-1.1-c1-0-187
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 + 4·13-s + 15-s − 4·17-s + 4·23-s + 25-s + 27-s − 6·29-s − 8·31-s + 33-s − 2·37-s + 4·39-s + 10·41-s − 10·43-s + 45-s − 12·47-s − 4·51-s + 6·53-s + 55-s − 12·59-s − 10·61-s + 4·65-s + 8·67-s + 4·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.258·15-s − 0.970·17-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s − 0.328·37-s + 0.640·39-s + 1.56·41-s − 1.52·43-s + 0.149·45-s − 1.75·47-s − 0.560·51-s + 0.824·53-s + 0.134·55-s − 1.56·59-s − 1.28·61-s + 0.496·65-s + 0.977·67-s + 0.481·69-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.66817421411510, −13.23175485312894, −12.93735884689697, −12.52065964082734, −11.67182242178497, −11.24437238622043, −10.87307011001510, −10.42965324413069, −9.648337219795697, −9.321889590332525, −8.857974198772175, −8.538402174705727, −7.853954367507165, −7.294342944884151, −6.855777119248534, −6.209108792860237, −5.887110966830923, −5.116543764388464, −4.623415126303920, −3.971114882321119, −3.378843628440901, −3.038591371919779, −1.980497705940092, −1.824587576943896, −1.003162616550832, 0, 1.003162616550832, 1.824587576943896, 1.980497705940092, 3.038591371919779, 3.378843628440901, 3.971114882321119, 4.623415126303920, 5.116543764388464, 5.887110966830923, 6.209108792860237, 6.855777119248534, 7.294342944884151, 7.853954367507165, 8.538402174705727, 8.857974198772175, 9.321889590332525, 9.648337219795697, 10.42965324413069, 10.87307011001510, 11.24437238622043, 11.67182242178497, 12.52065964082734, 12.93735884689697, 13.23175485312894, 13.66817421411510

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