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Properties

Label 2-117600-1.1-c1-0-130
Degree 22
Conductor 117600117600
Sign −1-1
Analytic cond. 939.040939.040
Root an. cond. 30.643730.6437
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 2·11-s − 3·17-s + 8·19-s − 7·23-s − 27-s − 2·29-s − 31-s − 2·33-s − 3·41-s + 8·43-s + 3·47-s + 3·51-s + 12·53-s − 8·57-s − 10·59-s − 2·61-s − 4·67-s + 7·69-s − 3·71-s − 2·73-s − 13·79-s + 81-s + 6·83-s + 2·87-s − 13·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.727·17-s + 1.83·19-s − 1.45·23-s − 0.192·27-s − 0.371·29-s − 0.179·31-s − 0.348·33-s − 0.468·41-s + 1.21·43-s + 0.437·47-s + 0.420·51-s + 1.64·53-s − 1.05·57-s − 1.30·59-s − 0.256·61-s − 0.488·67-s + 0.842·69-s − 0.356·71-s − 0.234·73-s − 1.46·79-s + 1/9·81-s + 0.658·83-s + 0.214·87-s − 1.37·89-s + ⋯

Functional equation

Λ(s)=(117600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(117600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117600117600    =    25⋅3⋅52⋅722^{5} \cdot 3 \cdot 5^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 939.040939.040
Root analytic conductor: 30.643730.6437
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 117600, ( :1/2), −1)(2,\ 117600,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+T 1 + T
5 1 1
7 1 1
good11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+7T+pT2 1 + 7 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1+pT2 1 + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+13T+pT2 1 + 13 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+13T+pT2 1 + 13 T + p T^{2}
97 1−9T+pT2 1 - 9 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−13.87568580669258, −13.35466378011667, −12.86275544380414, −12.17852242981210, −11.82939515939878, −11.61920133896993, −10.90925874655133, −10.49450520868943, −9.915815043259889, −9.476602260566418, −9.010490968684737, −8.476825339853128, −7.715595677851434, −7.371977625736020, −6.898902141734362, −6.198363574016130, −5.773455342104529, −5.390394652237799, −4.587916174080013, −4.186362270638373, −3.594300447857252, −2.939660494306464, −2.167224567001075, −1.517835619568320, −0.8456759066572029, 0, 0.8456759066572029, 1.517835619568320, 2.167224567001075, 2.939660494306464, 3.594300447857252, 4.186362270638373, 4.587916174080013, 5.390394652237799, 5.773455342104529, 6.198363574016130, 6.898902141734362, 7.371977625736020, 7.715595677851434, 8.476825339853128, 9.010490968684737, 9.476602260566418, 9.915815043259889, 10.49450520868943, 10.90925874655133, 11.61920133896993, 11.82939515939878, 12.17852242981210, 12.86275544380414, 13.35466378011667, 13.87568580669258

Graph of the ZZ-function along the critical line