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Properties

Label 2-4925-1.1-c1-0-121
Degree 22
Conductor 49254925
Sign −1-1
Analytic cond. 39.326339.3263
Root an. cond. 6.271076.27107
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  − 1.93·2-s − 0.393·3-s + 1.75·4-s + 0.762·6-s − 4.57·7-s + 0.481·8-s − 2.84·9-s − 2.65·11-s − 0.689·12-s + 6.10·13-s + 8.85·14-s − 4.43·16-s − 3.76·17-s + 5.51·18-s − 1.25·19-s + 1.79·21-s + 5.14·22-s + 8.79·23-s − 0.189·24-s − 11.8·26-s + 2.30·27-s − 8.00·28-s − 3.88·29-s − 4.36·31-s + 7.62·32-s + 1.04·33-s + 7.29·34-s + ⋯
L(s)  = 1  − 1.36·2-s − 0.227·3-s + 0.875·4-s + 0.311·6-s − 1.72·7-s + 0.170·8-s − 0.948·9-s − 0.801·11-s − 0.198·12-s + 1.69·13-s + 2.36·14-s − 1.10·16-s − 0.913·17-s + 1.29·18-s − 0.288·19-s + 0.392·21-s + 1.09·22-s + 1.83·23-s − 0.0387·24-s − 2.31·26-s + 0.442·27-s − 1.51·28-s − 0.720·29-s − 0.783·31-s + 1.34·32-s + 0.182·33-s + 1.25·34-s + ⋯

Functional equation

Λ(s)=(4925s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4925s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 49254925    =    52⋅1975^{2} \cdot 197
Sign: −1-1
Analytic conductor: 39.326339.3263
Root analytic conductor: 6.271076.27107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4925, ( :1/2), −1)(2,\ 4925,\ (\ :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)
bad5 1 1
197 1−T 1 - T
good2 1+1.93T+2T2 1 + 1.93T + 2T^{2}
3 1+0.393T+3T2 1 + 0.393T + 3T^{2}
7 1+4.57T+7T2 1 + 4.57T + 7T^{2}
11 1+2.65T+11T2 1 + 2.65T + 11T^{2}
13 1−6.10T+13T2 1 - 6.10T + 13T^{2}
17 1+3.76T+17T2 1 + 3.76T + 17T^{2}
19 1+1.25T+19T2 1 + 1.25T + 19T^{2}
23 1−8.79T+23T2 1 - 8.79T + 23T^{2}
29 1+3.88T+29T2 1 + 3.88T + 29T^{2}
31 1+4.36T+31T2 1 + 4.36T + 31T^{2}
37 1−2.74T+37T2 1 - 2.74T + 37T^{2}
41 1−2.33T+41T2 1 - 2.33T + 41T^{2}
43 1+6.44T+43T2 1 + 6.44T + 43T^{2}
47 1−0.0382T+47T2 1 - 0.0382T + 47T^{2}
53 1−5.36T+53T2 1 - 5.36T + 53T^{2}
59 1+4.00T+59T2 1 + 4.00T + 59T^{2}
61 1−14.5T+61T2 1 - 14.5T + 61T^{2}
67 1−13.5T+67T2 1 - 13.5T + 67T^{2}
71 1+0.772T+71T2 1 + 0.772T + 71T^{2}
73 1−11.9T+73T2 1 - 11.9T + 73T^{2}
79 1+14.2T+79T2 1 + 14.2T + 79T^{2}
83 1+10.2T+83T2 1 + 10.2T + 83T^{2}
89 1−5.99T+89T2 1 - 5.99T + 89T^{2}
97 1+12.8T+97T2 1 + 12.8T + 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

−8.226941250497167976600033414437, −7.11678377184883781104172684956, −6.68586192240100697051777816328, −5.96217595274541023465402628428, −5.19543602572875450510220208611, −3.91408077849314882809727692623, −3.12786162217130342047456352035, −2.28699540722660831141221637238, −0.878802522521460609329094736442, 0, 0.878802522521460609329094736442, 2.28699540722660831141221637238, 3.12786162217130342047456352035, 3.91408077849314882809727692623, 5.19543602572875450510220208611, 5.96217595274541023465402628428, 6.68586192240100697051777816328, 7.11678377184883781104172684956, 8.226941250497167976600033414437

Graph of the ZZ-function along the critical line