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Properties

Label 2-35e2-1.1-c1-0-14
Degree 22
Conductor 12251225
Sign 11
Analytic cond. 9.781679.78167
Root an. cond. 3.127563.12756
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.381·2-s + 0.874·3-s − 1.85·4-s + 0.333·6-s − 1.47·8-s − 2.23·9-s + 2.23·11-s − 1.62·12-s − 4.03·13-s + 3.14·16-s + 7.40·17-s − 0.854·18-s + 4.24·19-s + 0.854·22-s + 3.76·23-s − 1.28·24-s − 1.54·26-s − 4.57·27-s + 2.23·29-s + 6.86·31-s + 4.14·32-s + 1.95·33-s + 2.82·34-s + 4.14·36-s + 10.7·37-s + 1.62·38-s − 3.52·39-s + ⋯
L(s)  = 1  + 0.270·2-s + 0.504·3-s − 0.927·4-s + 0.136·6-s − 0.520·8-s − 0.745·9-s + 0.674·11-s − 0.467·12-s − 1.11·13-s + 0.786·16-s + 1.79·17-s − 0.201·18-s + 0.973·19-s + 0.182·22-s + 0.784·23-s − 0.262·24-s − 0.302·26-s − 0.880·27-s + 0.415·29-s + 1.23·31-s + 0.732·32-s + 0.340·33-s + 0.485·34-s + 0.690·36-s + 1.76·37-s + 0.262·38-s − 0.564·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 12251225    =    52⋅725^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 9.781679.78167
Root analytic conductor: 3.127563.12756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1225, ( :1/2), 1)(2,\ 1225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7235122861.723512286
L(12)L(\frac12) ≈\approx 1.7235122861.723512286
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)
bad5 1 1
7 1 1
good2 1−0.381T+2T2 1 - 0.381T + 2T^{2}
3 1−0.874T+3T2 1 - 0.874T + 3T^{2}
11 1−2.23T+11T2 1 - 2.23T + 11T^{2}
13 1+4.03T+13T2 1 + 4.03T + 13T^{2}
17 1−7.40T+17T2 1 - 7.40T + 17T^{2}
19 1−4.24T+19T2 1 - 4.24T + 19T^{2}
23 1−3.76T+23T2 1 - 3.76T + 23T^{2}
29 1−2.23T+29T2 1 - 2.23T + 29T^{2}
31 1−6.86T+31T2 1 - 6.86T + 31T^{2}
37 1−10.7T+37T2 1 - 10.7T + 37T^{2}
41 1+4.78T+41T2 1 + 4.78T + 41T^{2}
43 1−5T+43T2 1 - 5T + 43T^{2}
47 1+9.48T+47T2 1 + 9.48T + 47T^{2}
53 1−9.70T+53T2 1 - 9.70T + 53T^{2}
59 1+13.1T+59T2 1 + 13.1T + 59T^{2}
61 1+3.03T+61T2 1 + 3.03T + 61T^{2}
67 1−8.70T+67T2 1 - 8.70T + 67T^{2}
71 1+7.47T+71T2 1 + 7.47T + 71T^{2}
73 1−2.62T+73T2 1 - 2.62T + 73T^{2}
79 1+4.70T+79T2 1 + 4.70T + 79T^{2}
83 1−6.86T+83T2 1 - 6.86T + 83T^{2}
89 1−7.94T+89T2 1 - 7.94T + 89T^{2}
97 1+10.1T+97T2 1 + 10.1T + 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

−9.621240875598589495070815766931, −9.018522445632058270191866144420, −8.080359635468571264499353096271, −7.53562621692909125231051199261, −6.21653577053897407005030519307, −5.35456083259767857676768520788, −4.61772433744267444750778081059, −3.43927221409349693872136768565, −2.81326485793824964433691962308, −0.955988543724692389751162287094, 0.955988543724692389751162287094, 2.81326485793824964433691962308, 3.43927221409349693872136768565, 4.61772433744267444750778081059, 5.35456083259767857676768520788, 6.21653577053897407005030519307, 7.53562621692909125231051199261, 8.080359635468571264499353096271, 9.018522445632058270191866144420, 9.621240875598589495070815766931

Graph of the ZZ-function along the critical line