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Properties

Label 2-9600-1.1-c1-0-150
Degree 22
Conductor 96009600
Sign −1-1
Analytic cond. 76.656376.6563
Root an. cond. 8.755368.75536
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5.12·7-s + 9-s − 2·11-s − 5.12·13-s + 1.12·17-s − 5.12·19-s + 5.12·21-s − 5.12·23-s + 27-s − 8.24·29-s + 7.12·31-s − 2·33-s + 5.12·37-s − 5.12·39-s − 2·41-s − 6.24·43-s − 13.1·47-s + 19.2·49-s + 1.12·51-s − 10·53-s − 5.12·57-s − 6·59-s − 2·61-s + 5.12·63-s − 6.24·67-s − 5.12·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.93·7-s + 0.333·9-s − 0.603·11-s − 1.42·13-s + 0.272·17-s − 1.17·19-s + 1.11·21-s − 1.06·23-s + 0.192·27-s − 1.53·29-s + 1.27·31-s − 0.348·33-s + 0.842·37-s − 0.820·39-s − 0.312·41-s − 0.952·43-s − 1.91·47-s + 2.74·49-s + 0.157·51-s − 1.37·53-s − 0.678·57-s − 0.781·59-s − 0.256·61-s + 0.645·63-s − 0.763·67-s − 0.616·69-s + ⋯

Functional equation

Λ(s)=(9600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 96009600    =    27⋅3⋅522^{7} \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 76.656376.6563
Root analytic conductor: 8.755368.75536
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9600, ( :1/2), −1)(2,\ 9600,\ (\ :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
good7 1−5.12T+7T2 1 - 5.12T + 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1+5.12T+13T2 1 + 5.12T + 13T^{2}
17 1−1.12T+17T2 1 - 1.12T + 17T^{2}
19 1+5.12T+19T2 1 + 5.12T + 19T^{2}
23 1+5.12T+23T2 1 + 5.12T + 23T^{2}
29 1+8.24T+29T2 1 + 8.24T + 29T^{2}
31 1−7.12T+31T2 1 - 7.12T + 31T^{2}
37 1−5.12T+37T2 1 - 5.12T + 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1+6.24T+43T2 1 + 6.24T + 43T^{2}
47 1+13.1T+47T2 1 + 13.1T + 47T^{2}
53 1+10T+53T2 1 + 10T + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+6.24T+67T2 1 + 6.24T + 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1−4.24T+73T2 1 - 4.24T + 73T^{2}
79 1−4.87T+79T2 1 - 4.87T + 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+10T+97T2 1 + 10T + 97T^{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

−7.68000445049182951241916863412, −6.82216966543961239273177646666, −5.91044808789046285880545403631, −4.98285669183716440274428081484, −4.71794775404622438084136123861, −3.96122367147727515843844431394, −2.83438031636290181879678904414, −2.08206522742949308385468259978, −1.57809938123855289726341672185, 0, 1.57809938123855289726341672185, 2.08206522742949308385468259978, 2.83438031636290181879678904414, 3.96122367147727515843844431394, 4.71794775404622438084136123861, 4.98285669183716440274428081484, 5.91044808789046285880545403631, 6.82216966543961239273177646666, 7.68000445049182951241916863412

Graph of the ZZ-function along the critical line