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Properties

Label 2-35e2-1.1-c3-0-169
Degree 22
Conductor 12251225
Sign −1-1
Analytic cond. 72.277372.2773
Root an. cond. 8.501608.50160
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.87·2-s − 4.14·3-s + 15.7·4-s − 20.2·6-s + 37.6·8-s − 9.78·9-s + 36.9·11-s − 65.2·12-s − 61.3·13-s + 57.7·16-s − 44.8·17-s − 47.6·18-s + 139.·19-s + 180.·22-s − 217.·23-s − 156.·24-s − 298.·26-s + 152.·27-s − 33.8·29-s − 124.·31-s − 20.2·32-s − 153.·33-s − 218.·34-s − 153.·36-s − 237.·37-s + 680.·38-s + 254.·39-s + ⋯
L(s)  = 1  + 1.72·2-s − 0.798·3-s + 1.96·4-s − 1.37·6-s + 1.66·8-s − 0.362·9-s + 1.01·11-s − 1.57·12-s − 1.30·13-s + 0.902·16-s − 0.639·17-s − 0.624·18-s + 1.68·19-s + 1.74·22-s − 1.97·23-s − 1.33·24-s − 2.25·26-s + 1.08·27-s − 0.216·29-s − 0.720·31-s − 0.111·32-s − 0.809·33-s − 1.10·34-s − 0.712·36-s − 1.05·37-s + 2.90·38-s + 1.04·39-s + ⋯

Functional equation

Λ(s)=(1225s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1225s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12251225    =    52⋅725^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 72.277372.2773
Root analytic conductor: 8.501608.50160
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1225, ( :3/2), −1)(2,\ 1225,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad5 1 1
7 1 1
good2 1−4.87T+8T2 1 - 4.87T + 8T^{2}
3 1+4.14T+27T2 1 + 4.14T + 27T^{2}
11 1−36.9T+1.33e3T2 1 - 36.9T + 1.33e3T^{2}
13 1+61.3T+2.19e3T2 1 + 61.3T + 2.19e3T^{2}
17 1+44.8T+4.91e3T2 1 + 44.8T + 4.91e3T^{2}
19 1−139.T+6.85e3T2 1 - 139.T + 6.85e3T^{2}
23 1+217.T+1.21e4T2 1 + 217.T + 1.21e4T^{2}
29 1+33.8T+2.43e4T2 1 + 33.8T + 2.43e4T^{2}
31 1+124.T+2.97e4T2 1 + 124.T + 2.97e4T^{2}
37 1+237.T+5.06e4T2 1 + 237.T + 5.06e4T^{2}
41 1+195.T+6.89e4T2 1 + 195.T + 6.89e4T^{2}
43 1−343.T+7.95e4T2 1 - 343.T + 7.95e4T^{2}
47 1−16.8T+1.03e5T2 1 - 16.8T + 1.03e5T^{2}
53 1+346.T+1.48e5T2 1 + 346.T + 1.48e5T^{2}
59 1+135.T+2.05e5T2 1 + 135.T + 2.05e5T^{2}
61 1+490.T+2.26e5T2 1 + 490.T + 2.26e5T^{2}
67 1+477.T+3.00e5T2 1 + 477.T + 3.00e5T^{2}
71 1−45.2T+3.57e5T2 1 - 45.2T + 3.57e5T^{2}
73 1−100.T+3.89e5T2 1 - 100.T + 3.89e5T^{2}
79 1−880.T+4.93e5T2 1 - 880.T + 4.93e5T^{2}
83 1+1.15e3T+5.71e5T2 1 + 1.15e3T + 5.71e5T^{2}
89 1+619.T+7.04e5T2 1 + 619.T + 7.04e5T^{2}
97 1+231.T+9.12e5T2 1 + 231.T + 9.12e5T^{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

−9.067176368169110199107479587714, −7.66588133764104473851040105360, −6.90577747017518638876188848603, −6.10266817647913224128306225781, −5.47377215237084821810903600770, −4.74744492993242245056402390935, −3.88066246383256787242973696272, −2.90755977607247918896845824918, −1.76817491752824106549789722053, 0, 1.76817491752824106549789722053, 2.90755977607247918896845824918, 3.88066246383256787242973696272, 4.74744492993242245056402390935, 5.47377215237084821810903600770, 6.10266817647913224128306225781, 6.90577747017518638876188848603, 7.66588133764104473851040105360, 9.067176368169110199107479587714

Graph of the ZZ-function along the critical line