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Properties

Label 2-4925-1.1-c1-0-263
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  − 0.321·2-s + 2.86·3-s − 1.89·4-s − 0.920·6-s − 0.829·7-s + 1.25·8-s + 5.18·9-s − 1.61·11-s − 5.42·12-s − 3.76·13-s + 0.266·14-s + 3.38·16-s − 2.25·17-s − 1.66·18-s + 5.21·19-s − 2.37·21-s + 0.520·22-s + 0.268·23-s + 3.58·24-s + 1.21·26-s + 6.24·27-s + 1.57·28-s − 6.37·29-s + 0.201·31-s − 3.59·32-s − 4.62·33-s + 0.725·34-s + ⋯
L(s)  = 1  − 0.227·2-s + 1.65·3-s − 0.948·4-s − 0.375·6-s − 0.313·7-s + 0.443·8-s + 1.72·9-s − 0.487·11-s − 1.56·12-s − 1.04·13-s + 0.0713·14-s + 0.847·16-s − 0.547·17-s − 0.393·18-s + 1.19·19-s − 0.517·21-s + 0.110·22-s + 0.0559·23-s + 0.732·24-s + 0.237·26-s + 1.20·27-s + 0.297·28-s − 1.18·29-s + 0.0362·31-s − 0.636·32-s − 0.805·33-s + 0.124·34-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+0.321T+2T2 1 + 0.321T + 2T^{2}
3 1−2.86T+3T2 1 - 2.86T + 3T^{2}
7 1+0.829T+7T2 1 + 0.829T + 7T^{2}
11 1+1.61T+11T2 1 + 1.61T + 11T^{2}
13 1+3.76T+13T2 1 + 3.76T + 13T^{2}
17 1+2.25T+17T2 1 + 2.25T + 17T^{2}
19 1−5.21T+19T2 1 - 5.21T + 19T^{2}
23 1−0.268T+23T2 1 - 0.268T + 23T^{2}
29 1+6.37T+29T2 1 + 6.37T + 29T^{2}
31 1−0.201T+31T2 1 - 0.201T + 31T^{2}
37 1+7.50T+37T2 1 + 7.50T + 37T^{2}
41 1−2.01T+41T2 1 - 2.01T + 41T^{2}
43 1+8.49T+43T2 1 + 8.49T + 43T^{2}
47 1−1.06T+47T2 1 - 1.06T + 47T^{2}
53 1−10.6T+53T2 1 - 10.6T + 53T^{2}
59 1+12.4T+59T2 1 + 12.4T + 59T^{2}
61 1−1.31T+61T2 1 - 1.31T + 61T^{2}
67 1+2.57T+67T2 1 + 2.57T + 67T^{2}
71 1−6.89T+71T2 1 - 6.89T + 71T^{2}
73 1−4.02T+73T2 1 - 4.02T + 73T^{2}
79 1−12.1T+79T2 1 - 12.1T + 79T^{2}
83 1+7.56T+83T2 1 + 7.56T + 83T^{2}
89 1+8.82T+89T2 1 + 8.82T + 89T^{2}
97 1−1.91T+97T2 1 - 1.91T + 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

−7.937881611201998708239402733640, −7.52901608853947672807117139903, −6.81483201885149868803583984628, −5.44589242574628705027085981739, −4.86549821908378579745718996839, −3.92132254390429415216292343705, −3.31103747073471707301262430071, −2.51732644497890736495429499493, −1.54472551587583614713312110723, 0, 1.54472551587583614713312110723, 2.51732644497890736495429499493, 3.31103747073471707301262430071, 3.92132254390429415216292343705, 4.86549821908378579745718996839, 5.44589242574628705027085981739, 6.81483201885149868803583984628, 7.52901608853947672807117139903, 7.937881611201998708239402733640

Graph of the ZZ-function along the critical line