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Properties

Label 2-650-1.1-c1-0-12
Degree 22
Conductor 650650
Sign 11
Analytic cond. 5.190275.19027
Root an. cond. 2.278212.27821
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  + 2-s + 2.29·3-s + 4-s + 2.29·6-s − 1.25·7-s + 8-s + 2.25·9-s + 2·11-s + 2.29·12-s + 13-s − 1.25·14-s + 16-s + 4.80·17-s + 2.25·18-s − 5.09·19-s − 2.87·21-s + 2·22-s − 2.58·23-s + 2.29·24-s + 26-s − 1.70·27-s − 1.25·28-s + 5.09·29-s + 8.58·31-s + 32-s + 4.58·33-s + 4.80·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.935·6-s − 0.474·7-s + 0.353·8-s + 0.751·9-s + 0.603·11-s + 0.661·12-s + 0.277·13-s − 0.335·14-s + 0.250·16-s + 1.16·17-s + 0.531·18-s − 1.16·19-s − 0.627·21-s + 0.426·22-s − 0.538·23-s + 0.467·24-s + 0.196·26-s − 0.328·27-s − 0.237·28-s + 0.946·29-s + 1.54·31-s + 0.176·32-s + 0.798·33-s + 0.823·34-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: 11
Analytic conductor: 5.190275.19027
Root analytic conductor: 2.278212.27821
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 650, ( :1/2), 1)(2,\ 650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3429169233.342916923
L(12)L(\frac12) ≈\approx 3.3429169233.342916923
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)
bad2 1−T 1 - T
5 1 1
13 1−T 1 - T
good3 1−2.29T+3T2 1 - 2.29T + 3T^{2}
7 1+1.25T+7T2 1 + 1.25T + 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
17 1−4.80T+17T2 1 - 4.80T + 17T^{2}
19 1+5.09T+19T2 1 + 5.09T + 19T^{2}
23 1+2.58T+23T2 1 + 2.58T + 23T^{2}
29 1−5.09T+29T2 1 - 5.09T + 29T^{2}
31 1−8.58T+31T2 1 - 8.58T + 31T^{2}
37 1+7.83T+37T2 1 + 7.83T + 37T^{2}
41 1+9.67T+41T2 1 + 9.67T + 41T^{2}
43 1+10.8T+43T2 1 + 10.8T + 43T^{2}
47 1+2.74T+47T2 1 + 2.74T + 47T^{2}
53 1+2.58T+53T2 1 + 2.58T + 53T^{2}
59 1+5.09T+59T2 1 + 5.09T + 59T^{2}
61 1−13.6T+61T2 1 - 13.6T + 61T^{2}
67 1+8.58T+67T2 1 + 8.58T + 67T^{2}
71 1−5.38T+71T2 1 - 5.38T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1−15.0T+79T2 1 - 15.0T + 79T^{2}
83 1+11.0T+83T2 1 + 11.0T + 83T^{2}
89 1−5.09T+89T2 1 - 5.09T + 89T^{2}
97 1−6.26T+97T2 1 - 6.26T + 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

−10.31070904435950212812600412734, −9.780154197229813220997409135135, −8.551098722056452767692876897214, −8.170853712915319883613391038322, −6.89503630605462440704286405668, −6.18108686749386443483918385014, −4.82216061412183189967938018675, −3.67191952564387840603540066156, −3.07107705661602441580678764547, −1.79092080666297334905946892235, 1.79092080666297334905946892235, 3.07107705661602441580678764547, 3.67191952564387840603540066156, 4.82216061412183189967938018675, 6.18108686749386443483918385014, 6.89503630605462440704286405668, 8.170853712915319883613391038322, 8.551098722056452767692876897214, 9.780154197229813220997409135135, 10.31070904435950212812600412734

Graph of the ZZ-function along the critical line