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Properties

Label 2-8512-1.1-c1-0-198
Degree 22
Conductor 85128512
Sign −1-1
Analytic cond. 67.968667.9686
Root an. cond. 8.244318.24431
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  + 2.69·3-s − 1.42·5-s + 7-s + 4.27·9-s − 3.27·11-s − 3.42·13-s − 3.84·15-s + 5.27·17-s − 19-s + 2.69·21-s − 0.574·23-s − 2.96·25-s + 3.42·27-s + 0.122·29-s − 2.12·31-s − 8.81·33-s − 1.42·35-s + 3.96·37-s − 9.23·39-s − 4.66·41-s − 11.9·43-s − 6.08·45-s − 3.11·47-s + 49-s + 14.2·51-s + 6.08·53-s + 4.66·55-s + ⋯
L(s)  = 1  + 1.55·3-s − 0.637·5-s + 0.377·7-s + 1.42·9-s − 0.986·11-s − 0.950·13-s − 0.992·15-s + 1.27·17-s − 0.229·19-s + 0.588·21-s − 0.119·23-s − 0.593·25-s + 0.659·27-s + 0.0227·29-s − 0.381·31-s − 1.53·33-s − 0.241·35-s + 0.652·37-s − 1.47·39-s − 0.728·41-s − 1.81·43-s − 0.907·45-s − 0.454·47-s + 0.142·49-s + 1.98·51-s + 0.836·53-s + 0.628·55-s + ⋯

Functional equation

Λ(s)=(8512s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8512s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 85128512    =    26⋅7⋅192^{6} \cdot 7 \cdot 19
Sign: −1-1
Analytic conductor: 67.968667.9686
Root analytic conductor: 8.244318.24431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8512, ( :1/2), −1)(2,\ 8512,\ (\ :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−T 1 - T
19 1+T 1 + T
good3 1−2.69T+3T2 1 - 2.69T + 3T^{2}
5 1+1.42T+5T2 1 + 1.42T + 5T^{2}
11 1+3.27T+11T2 1 + 3.27T + 11T^{2}
13 1+3.42T+13T2 1 + 3.42T + 13T^{2}
17 1−5.27T+17T2 1 - 5.27T + 17T^{2}
23 1+0.574T+23T2 1 + 0.574T + 23T^{2}
29 1−0.122T+29T2 1 - 0.122T + 29T^{2}
31 1+2.12T+31T2 1 + 2.12T + 31T^{2}
37 1−3.96T+37T2 1 - 3.96T + 37T^{2}
41 1+4.66T+41T2 1 + 4.66T + 41T^{2}
43 1+11.9T+43T2 1 + 11.9T + 43T^{2}
47 1+3.11T+47T2 1 + 3.11T + 47T^{2}
53 1−6.08T+53T2 1 - 6.08T + 53T^{2}
59 1+1.72T+59T2 1 + 1.72T + 59T^{2}
61 1+12.2T+61T2 1 + 12.2T + 61T^{2}
67 1+5.27T+67T2 1 + 5.27T + 67T^{2}
71 1−6.81T+71T2 1 - 6.81T + 71T^{2}
73 1+11.5T+73T2 1 + 11.5T + 73T^{2}
79 1+11.0T+79T2 1 + 11.0T + 79T^{2}
83 1−5.23T+83T2 1 - 5.23T + 83T^{2}
89 1+1.45T+89T2 1 + 1.45T + 89T^{2}
97 1−17.6T+97T2 1 - 17.6T + 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

−7.49335196015445891169663774415, −7.32917433019601515780686071811, −6.06779315280294485683854622066, −5.14930638794445310424244747070, −4.54055392196831253313043191472, −3.63034305755035502871364709165, −3.10860984009089974773313640115, −2.36084623796459330576927140568, −1.55043815102810441727570080043, 0, 1.55043815102810441727570080043, 2.36084623796459330576927140568, 3.10860984009089974773313640115, 3.63034305755035502871364709165, 4.54055392196831253313043191472, 5.14930638794445310424244747070, 6.06779315280294485683854622066, 7.32917433019601515780686071811, 7.49335196015445891169663774415

Graph of the ZZ-function along the critical line