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Properties

Label 2-326700-1.1-c1-0-129
Degree 22
Conductor 326700326700
Sign −1-1
Analytic cond. 2608.712608.71
Root an. cond. 51.075551.0755
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  − 4·7-s + 5·13-s + 19-s + 11·31-s + 37-s + 8·43-s + 9·49-s + 13·61-s + 16·67-s − 10·73-s + 4·79-s − 20·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.38·13-s + 0.229·19-s + 1.97·31-s + 0.164·37-s + 1.21·43-s + 9/7·49-s + 1.66·61-s + 1.95·67-s − 1.17·73-s + 0.450·79-s − 2.09·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

Λ(s)=(326700s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 326700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(326700s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 326700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 326700326700    =    22⋅33⋅52⋅1122^{2} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 2608.712608.71
Root analytic conductor: 51.075551.0755
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 326700, ( :1/2), −1)(2,\ 326700,\ (\ :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 1
5 1 1
11 1 1
good7 1+4T+pT2 1 + 4 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−11T+pT2 1 - 11 T + p T^{2}
37 1−T+pT2 1 - T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−13T+pT2 1 - 13 T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+14T+pT2 1 + 14 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

−12.86250162851920, −12.47586797319510, −11.98333139634806, −11.39257427347417, −11.11810382485074, −10.43279299137437, −10.10870869260594, −9.662399453633723, −9.244544754458528, −8.728360265439167, −8.266236263648501, −7.877076851123835, −7.114958539823117, −6.658043207089988, −6.429191573785297, −5.798770618997876, −5.548811141825750, −4.715004435006800, −4.140847300579289, −3.701347468387505, −3.244640774753684, −2.672966075532914, −2.227173113762841, −1.149322284211527, −0.8807505699413323, 0, 0.8807505699413323, 1.149322284211527, 2.227173113762841, 2.672966075532914, 3.244640774753684, 3.701347468387505, 4.140847300579289, 4.715004435006800, 5.548811141825750, 5.798770618997876, 6.429191573785297, 6.658043207089988, 7.114958539823117, 7.877076851123835, 8.266236263648501, 8.728360265439167, 9.244544754458528, 9.662399453633723, 10.10870869260594, 10.43279299137437, 11.11810382485074, 11.39257427347417, 11.98333139634806, 12.47586797319510, 12.86250162851920

Graph of the ZZ-function along the critical line