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Properties

Label 2-5175-1.1-c1-0-76
Degree 22
Conductor 51755175
Sign 11
Analytic cond. 41.322541.3225
Root an. cond. 6.428266.42826
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  + 0.161·2-s − 1.97·4-s + 5.15·7-s − 0.641·8-s + 1.28·11-s + 1.53·13-s + 0.832·14-s + 3.84·16-s + 2.43·17-s + 6.13·19-s + 0.207·22-s + 23-s + 0.247·26-s − 10.1·28-s + 6.10·29-s + 0.248·31-s + 1.90·32-s + 0.392·34-s − 1.29·37-s + 0.991·38-s − 5.44·41-s − 5.64·43-s − 2.53·44-s + 0.161·46-s + 7.37·47-s + 19.5·49-s − 3.02·52-s + ⋯
L(s)  = 1  + 0.114·2-s − 0.986·4-s + 1.94·7-s − 0.226·8-s + 0.387·11-s + 0.424·13-s + 0.222·14-s + 0.961·16-s + 0.590·17-s + 1.40·19-s + 0.0442·22-s + 0.208·23-s + 0.0485·26-s − 1.92·28-s + 1.13·29-s + 0.0445·31-s + 0.336·32-s + 0.0673·34-s − 0.213·37-s + 0.160·38-s − 0.849·41-s − 0.861·43-s − 0.382·44-s + 0.0238·46-s + 1.07·47-s + 2.79·49-s − 0.419·52-s + ⋯

Functional equation

Λ(s)=(5175s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5175s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 51755175    =    32⋅52⋅233^{2} \cdot 5^{2} \cdot 23
Sign: 11
Analytic conductor: 41.322541.3225
Root analytic conductor: 6.428266.42826
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5175, ( :1/2), 1)(2,\ 5175,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4874582632.487458263
L(12)L(\frac12) ≈\approx 2.4874582632.487458263
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)
bad3 1 1
5 1 1
23 1−T 1 - T
good2 1−0.161T+2T2 1 - 0.161T + 2T^{2}
7 1−5.15T+7T2 1 - 5.15T + 7T^{2}
11 1−1.28T+11T2 1 - 1.28T + 11T^{2}
13 1−1.53T+13T2 1 - 1.53T + 13T^{2}
17 1−2.43T+17T2 1 - 2.43T + 17T^{2}
19 1−6.13T+19T2 1 - 6.13T + 19T^{2}
29 1−6.10T+29T2 1 - 6.10T + 29T^{2}
31 1−0.248T+31T2 1 - 0.248T + 31T^{2}
37 1+1.29T+37T2 1 + 1.29T + 37T^{2}
41 1+5.44T+41T2 1 + 5.44T + 41T^{2}
43 1+5.64T+43T2 1 + 5.64T + 43T^{2}
47 1−7.37T+47T2 1 - 7.37T + 47T^{2}
53 1+7.36T+53T2 1 + 7.36T + 53T^{2}
59 1+12.9T+59T2 1 + 12.9T + 59T^{2}
61 1−10.3T+61T2 1 - 10.3T + 61T^{2}
67 1+5.75T+67T2 1 + 5.75T + 67T^{2}
71 1−10.2T+71T2 1 - 10.2T + 71T^{2}
73 1−5.51T+73T2 1 - 5.51T + 73T^{2}
79 1−0.0974T+79T2 1 - 0.0974T + 79T^{2}
83 1+8.84T+83T2 1 + 8.84T + 83T^{2}
89 1+16.6T+89T2 1 + 16.6T + 89T^{2}
97 1−9.68T+97T2 1 - 9.68T + 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

−8.180571160789384477489724100609, −7.78291819230239062596326243141, −6.87256200835602930706097462469, −5.73641157115380474297566311279, −5.16805918221033122960261224255, −4.65984663596363344337023922391, −3.87786113472940641447052278137, −2.99330207757845276578569083605, −1.58807710115916272239700248441, −0.970801217109611142879721505640, 0.970801217109611142879721505640, 1.58807710115916272239700248441, 2.99330207757845276578569083605, 3.87786113472940641447052278137, 4.65984663596363344337023922391, 5.16805918221033122960261224255, 5.73641157115380474297566311279, 6.87256200835602930706097462469, 7.78291819230239062596326243141, 8.180571160789384477489724100609

Graph of the ZZ-function along the critical line