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Properties

Label 2-129360-1.1-c1-0-122
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 + 6·23-s + 25-s − 27-s − 10·29-s − 2·31-s + 33-s + 2·37-s − 2·39-s − 8·41-s − 45-s − 12·47-s − 2·51-s + 2·53-s + 55-s − 4·57-s − 10·59-s + 4·61-s − 2·65-s + 10·67-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 + 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.359·31-s + 0.174·33-s + 0.328·37-s − 0.320·39-s − 1.24·41-s − 0.149·45-s − 1.75·47-s − 0.280·51-s + 0.274·53-s + 0.134·55-s − 0.529·57-s − 1.30·59-s + 0.512·61-s − 0.248·65-s + 1.22·67-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 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.57669587964143, −13.17702048493059, −12.78518183086073, −12.30639145371356, −11.63530221390933, −11.33985673227441, −11.02707941753917, −10.39751797857210, −9.924420824971994, −9.269157493119544, −9.051544705592494, −8.097878351227421, −7.950798651387293, −7.204042542362067, −6.912238650857425, −6.238544635575735, −5.660360849776661, −5.133047454189603, −4.858126661080722, −3.994228507369785, −3.403865464715014, −3.156679036062716, −2.118629027676407, −1.467884045941417, −0.8000747126012352, 0, 0.8000747126012352, 1.467884045941417, 2.118629027676407, 3.156679036062716, 3.403865464715014, 3.994228507369785, 4.858126661080722, 5.133047454189603, 5.660360849776661, 6.238544635575735, 6.912238650857425, 7.204042542362067, 7.950798651387293, 8.097878351227421, 9.051544705592494, 9.269157493119544, 9.924420824971994, 10.39751797857210, 11.02707941753917, 11.33985673227441, 11.63530221390933, 12.30639145371356, 12.78518183086073, 13.17702048493059, 13.57669587964143

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