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Properties

Label 2-35e2-1.1-c3-0-9
Degree 22
Conductor 12251225
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.915·2-s − 8.77·3-s − 7.16·4-s + 8.03·6-s + 13.8·8-s + 49.9·9-s − 45.6·11-s + 62.8·12-s − 16.0·13-s + 44.5·16-s + 7.28·17-s − 45.7·18-s + 75.6·19-s + 41.8·22-s − 15.8·23-s − 121.·24-s + 14.7·26-s − 201.·27-s + 119.·29-s + 59.5·31-s − 151.·32-s + 400.·33-s − 6.67·34-s − 357.·36-s − 304.·37-s − 69.2·38-s + 141.·39-s + ⋯
L(s)  = 1  − 0.323·2-s − 1.68·3-s − 0.895·4-s + 0.546·6-s + 0.613·8-s + 1.85·9-s − 1.25·11-s + 1.51·12-s − 0.343·13-s + 0.696·16-s + 0.103·17-s − 0.599·18-s + 0.913·19-s + 0.405·22-s − 0.143·23-s − 1.03·24-s + 0.111·26-s − 1.43·27-s + 0.764·29-s + 0.345·31-s − 0.839·32-s + 2.11·33-s − 0.0336·34-s − 1.65·36-s − 1.35·37-s − 0.295·38-s + 0.579·39-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: 11
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: 00
Selberg data: (2, 1225, ( :3/2), 1)(2,\ 1225,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.25537081120.2553708112
L(12)L(\frac12) ≈\approx 0.25537081120.2553708112
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+0.915T+8T2 1 + 0.915T + 8T^{2}
3 1+8.77T+27T2 1 + 8.77T + 27T^{2}
11 1+45.6T+1.33e3T2 1 + 45.6T + 1.33e3T^{2}
13 1+16.0T+2.19e3T2 1 + 16.0T + 2.19e3T^{2}
17 1−7.28T+4.91e3T2 1 - 7.28T + 4.91e3T^{2}
19 1−75.6T+6.85e3T2 1 - 75.6T + 6.85e3T^{2}
23 1+15.8T+1.21e4T2 1 + 15.8T + 1.21e4T^{2}
29 1−119.T+2.43e4T2 1 - 119.T + 2.43e4T^{2}
31 1−59.5T+2.97e4T2 1 - 59.5T + 2.97e4T^{2}
37 1+304.T+5.06e4T2 1 + 304.T + 5.06e4T^{2}
41 1+238.T+6.89e4T2 1 + 238.T + 6.89e4T^{2}
43 1+365.T+7.95e4T2 1 + 365.T + 7.95e4T^{2}
47 1+100.T+1.03e5T2 1 + 100.T + 1.03e5T^{2}
53 1+740.T+1.48e5T2 1 + 740.T + 1.48e5T^{2}
59 1−285.T+2.05e5T2 1 - 285.T + 2.05e5T^{2}
61 1+400.T+2.26e5T2 1 + 400.T + 2.26e5T^{2}
67 1−128.T+3.00e5T2 1 - 128.T + 3.00e5T^{2}
71 1+39.1T+3.57e5T2 1 + 39.1T + 3.57e5T^{2}
73 1+961.T+3.89e5T2 1 + 961.T + 3.89e5T^{2}
79 1+552.T+4.93e5T2 1 + 552.T + 4.93e5T^{2}
83 1−33.6T+5.71e5T2 1 - 33.6T + 5.71e5T^{2}
89 1−856.T+7.04e5T2 1 - 856.T + 7.04e5T^{2}
97 1−771.T+9.12e5T2 1 - 771.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

−9.670596033934037869558627683029, −8.488537605778158335548270284079, −7.68262039875955515929687413573, −6.82797063036917393026725702417, −5.79068778318718189288028155772, −5.04598011662329551835821072085, −4.69060612358520639276087991973, −3.26906649466948532804918315785, −1.47812967960477643445960167529, −0.31531534801067252471518172766, 0.31531534801067252471518172766, 1.47812967960477643445960167529, 3.26906649466948532804918315785, 4.69060612358520639276087991973, 5.04598011662329551835821072085, 5.79068778318718189288028155772, 6.82797063036917393026725702417, 7.68262039875955515929687413573, 8.488537605778158335548270284079, 9.670596033934037869558627683029

Graph of the ZZ-function along the critical line