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Properties

Label 2-8624-1.1-c1-0-194
Degree 22
Conductor 86248624
Sign −1-1
Analytic cond. 68.862968.8629
Root an. cond. 8.298378.29837
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  + 2.82·3-s − 1.41·5-s + 5.00·9-s + 11-s + 1.41·13-s − 4.00·15-s − 7.07·17-s + 2.82·19-s − 4·23-s − 2.99·25-s + 5.65·27-s − 5.65·31-s + 2.82·33-s − 8·37-s + 4.00·39-s − 9.89·41-s − 4·43-s − 7.07·45-s − 20.0·51-s − 6·53-s − 1.41·55-s + 8.00·57-s + 8.48·59-s + 1.41·61-s − 2.00·65-s + 8·67-s − 11.3·69-s + ⋯
L(s)  = 1  + 1.63·3-s − 0.632·5-s + 1.66·9-s + 0.301·11-s + 0.392·13-s − 1.03·15-s − 1.71·17-s + 0.648·19-s − 0.834·23-s − 0.599·25-s + 1.08·27-s − 1.01·31-s + 0.492·33-s − 1.31·37-s + 0.640·39-s − 1.54·41-s − 0.609·43-s − 1.05·45-s − 2.80·51-s − 0.824·53-s − 0.190·55-s + 1.05·57-s + 1.10·59-s + 0.181·61-s − 0.248·65-s + 0.977·67-s − 1.36·69-s + ⋯

Functional equation

Λ(s)=(8624s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8624s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 86248624    =    24⋅72⋅112^{4} \cdot 7^{2} \cdot 11
Sign: −1-1
Analytic conductor: 68.862968.8629
Root analytic conductor: 8.298378.29837
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8624, ( :1/2), −1)(2,\ 8624,\ (\ :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
7 1 1
11 1−T 1 - T
good3 1−2.82T+3T2 1 - 2.82T + 3T^{2}
5 1+1.41T+5T2 1 + 1.41T + 5T^{2}
13 1−1.41T+13T2 1 - 1.41T + 13T^{2}
17 1+7.07T+17T2 1 + 7.07T + 17T^{2}
19 1−2.82T+19T2 1 - 2.82T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1+5.65T+31T2 1 + 5.65T + 31T^{2}
37 1+8T+37T2 1 + 8T + 37T^{2}
41 1+9.89T+41T2 1 + 9.89T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1−8.48T+59T2 1 - 8.48T + 59T^{2}
61 1−1.41T+61T2 1 - 1.41T + 61T^{2}
67 1−8T+67T2 1 - 8T + 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1−1.41T+73T2 1 - 1.41T + 73T^{2}
79 1+16T+79T2 1 + 16T + 79T^{2}
83 1+2.82T+83T2 1 + 2.82T + 83T^{2}
89 1−15.5T+89T2 1 - 15.5T + 89T^{2}
97 1+9.89T+97T2 1 + 9.89T + 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

−7.46760761991361247988733304208, −7.04413911466501462583926811631, −6.26009793297953143686702153148, −5.17855580284254959492776254623, −4.29657086124467134879326308352, −3.70980917516283042942945851155, −3.23965352148593878850875245520, −2.18229187698172178580742072031, −1.64898519759039438656792173194, 0, 1.64898519759039438656792173194, 2.18229187698172178580742072031, 3.23965352148593878850875245520, 3.70980917516283042942945851155, 4.29657086124467134879326308352, 5.17855580284254959492776254623, 6.26009793297953143686702153148, 7.04413911466501462583926811631, 7.46760761991361247988733304208

Graph of the ZZ-function along the critical line