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Properties

Label 2-4925-1.1-c1-0-133
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  + 0.209·2-s − 1.22·3-s − 1.95·4-s − 0.258·6-s − 2.44·7-s − 0.830·8-s − 1.48·9-s − 2.39·11-s + 2.40·12-s − 0.190·13-s − 0.512·14-s + 3.73·16-s + 5.21·17-s − 0.312·18-s + 1.40·19-s + 3.00·21-s − 0.502·22-s − 5.67·23-s + 1.02·24-s − 0.0400·26-s + 5.51·27-s + 4.78·28-s + 8.18·29-s − 1.99·31-s + 2.44·32-s + 2.94·33-s + 1.09·34-s + ⋯
L(s)  = 1  + 0.148·2-s − 0.710·3-s − 0.977·4-s − 0.105·6-s − 0.923·7-s − 0.293·8-s − 0.495·9-s − 0.721·11-s + 0.694·12-s − 0.0529·13-s − 0.137·14-s + 0.934·16-s + 1.26·17-s − 0.0735·18-s + 0.322·19-s + 0.656·21-s − 0.107·22-s − 1.18·23-s + 0.208·24-s − 0.00785·26-s + 1.06·27-s + 0.903·28-s + 1.51·29-s − 0.358·31-s + 0.432·32-s + 0.512·33-s + 0.187·34-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 1−0.209T+2T2 1 - 0.209T + 2T^{2}
3 1+1.22T+3T2 1 + 1.22T + 3T^{2}
7 1+2.44T+7T2 1 + 2.44T + 7T^{2}
11 1+2.39T+11T2 1 + 2.39T + 11T^{2}
13 1+0.190T+13T2 1 + 0.190T + 13T^{2}
17 1−5.21T+17T2 1 - 5.21T + 17T^{2}
19 1−1.40T+19T2 1 - 1.40T + 19T^{2}
23 1+5.67T+23T2 1 + 5.67T + 23T^{2}
29 1−8.18T+29T2 1 - 8.18T + 29T^{2}
31 1+1.99T+31T2 1 + 1.99T + 31T^{2}
37 1−7.49T+37T2 1 - 7.49T + 37T^{2}
41 1+5.17T+41T2 1 + 5.17T + 41T^{2}
43 1+3.60T+43T2 1 + 3.60T + 43T^{2}
47 1−0.599T+47T2 1 - 0.599T + 47T^{2}
53 1−10.1T+53T2 1 - 10.1T + 53T^{2}
59 1−3.88T+59T2 1 - 3.88T + 59T^{2}
61 1−12.5T+61T2 1 - 12.5T + 61T^{2}
67 1−1.88T+67T2 1 - 1.88T + 67T^{2}
71 1+3.97T+71T2 1 + 3.97T + 71T^{2}
73 1−3.17T+73T2 1 - 3.17T + 73T^{2}
79 1+7.95T+79T2 1 + 7.95T + 79T^{2}
83 1+8.39T+83T2 1 + 8.39T + 83T^{2}
89 1+6.48T+89T2 1 + 6.48T + 89T^{2}
97 1−8.97T+97T2 1 - 8.97T + 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

−8.136318057991017884120372876681, −7.08262591622937181668435882072, −6.21581855049938004139597044590, −5.60930030857261300108653238946, −5.16155892505088371188084739291, −4.21113375255676152658357744908, −3.36265629060283159304357206564, −2.65531691259051274139482461074, −0.941982740346138751930070943118, 0, 0.941982740346138751930070943118, 2.65531691259051274139482461074, 3.36265629060283159304357206564, 4.21113375255676152658357744908, 5.16155892505088371188084739291, 5.60930030857261300108653238946, 6.21581855049938004139597044590, 7.08262591622937181668435882072, 8.136318057991017884120372876681

Graph of the ZZ-function along the critical line