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Properties

Label 2-4925-1.1-c1-0-126
Degree 22
Conductor 49254925
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.70·2-s − 0.982·3-s + 0.895·4-s + 1.67·6-s + 5.03·7-s + 1.87·8-s − 2.03·9-s + 2.51·11-s − 0.880·12-s + 5.29·13-s − 8.56·14-s − 4.98·16-s − 5.08·17-s + 3.46·18-s + 3.09·19-s − 4.94·21-s − 4.28·22-s + 7.49·23-s − 1.84·24-s − 9.01·26-s + 4.94·27-s + 4.51·28-s + 8.95·29-s − 4.27·31-s + 4.73·32-s − 2.47·33-s + 8.65·34-s + ⋯
L(s)  = 1  − 1.20·2-s − 0.567·3-s + 0.447·4-s + 0.682·6-s + 1.90·7-s + 0.664·8-s − 0.678·9-s + 0.759·11-s − 0.254·12-s + 1.46·13-s − 2.28·14-s − 1.24·16-s − 1.23·17-s + 0.816·18-s + 0.709·19-s − 1.07·21-s − 0.913·22-s + 1.56·23-s − 0.376·24-s − 1.76·26-s + 0.952·27-s + 0.852·28-s + 1.66·29-s − 0.767·31-s + 0.836·32-s − 0.430·33-s + 1.48·34-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: 11
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: 00
Selberg data: (2, 4925, ( :1/2), 1)(2,\ 4925,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2165631741.216563174
L(12)L(\frac12) ≈\approx 1.2165631741.216563174
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+1.70T+2T2 1 + 1.70T + 2T^{2}
3 1+0.982T+3T2 1 + 0.982T + 3T^{2}
7 1−5.03T+7T2 1 - 5.03T + 7T^{2}
11 1−2.51T+11T2 1 - 2.51T + 11T^{2}
13 1−5.29T+13T2 1 - 5.29T + 13T^{2}
17 1+5.08T+17T2 1 + 5.08T + 17T^{2}
19 1−3.09T+19T2 1 - 3.09T + 19T^{2}
23 1−7.49T+23T2 1 - 7.49T + 23T^{2}
29 1−8.95T+29T2 1 - 8.95T + 29T^{2}
31 1+4.27T+31T2 1 + 4.27T + 31T^{2}
37 1−0.741T+37T2 1 - 0.741T + 37T^{2}
41 1+1.33T+41T2 1 + 1.33T + 41T^{2}
43 1−9.49T+43T2 1 - 9.49T + 43T^{2}
47 1+2.68T+47T2 1 + 2.68T + 47T^{2}
53 1−1.33T+53T2 1 - 1.33T + 53T^{2}
59 1−0.910T+59T2 1 - 0.910T + 59T^{2}
61 1+5.51T+61T2 1 + 5.51T + 61T^{2}
67 1−2.79T+67T2 1 - 2.79T + 67T^{2}
71 1−13.3T+71T2 1 - 13.3T + 71T^{2}
73 1−12.8T+73T2 1 - 12.8T + 73T^{2}
79 1+6.18T+79T2 1 + 6.18T + 79T^{2}
83 1+11.9T+83T2 1 + 11.9T + 83T^{2}
89 1−3.68T+89T2 1 - 3.68T + 89T^{2}
97 1−8.00T+97T2 1 - 8.00T + 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

−8.514018134746900192816231540606, −7.78117234444899395448261220078, −6.94979725431502884073650500967, −6.26785582527633759606756079011, −5.23195128053892604489415183051, −4.74174326145237699721794320085, −3.89038229473508573528557701469, −2.48523006200615856326536295270, −1.36020398260320457395866743069, −0.899244788065390005946172557163, 0.899244788065390005946172557163, 1.36020398260320457395866743069, 2.48523006200615856326536295270, 3.89038229473508573528557701469, 4.74174326145237699721794320085, 5.23195128053892604489415183051, 6.26785582527633759606756079011, 6.94979725431502884073650500967, 7.78117234444899395448261220078, 8.514018134746900192816231540606

Graph of the ZZ-function along the critical line