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Properties

Label 2-4650-1.1-c1-0-13
Degree 22
Conductor 46504650
Sign 11
Analytic cond. 37.130437.1304
Root an. cond. 6.093476.09347
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 − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s + 3.26·11-s − 12-s − 2.73·13-s + 2·14-s + 16-s + 4.93·17-s − 18-s + 4.93·19-s + 2·21-s − 3.26·22-s + 0.521·23-s + 24-s + 2.73·26-s − 27-s − 2·28-s + 0.521·29-s − 31-s − 32-s − 3.26·33-s − 4.93·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.333·9-s + 0.983·11-s − 0.288·12-s − 0.759·13-s + 0.534·14-s + 0.250·16-s + 1.19·17-s − 0.235·18-s + 1.13·19-s + 0.436·21-s − 0.695·22-s + 0.108·23-s + 0.204·24-s + 0.537·26-s − 0.192·27-s − 0.377·28-s + 0.0968·29-s − 0.179·31-s − 0.176·32-s − 0.567·33-s − 0.845·34-s + ⋯

Functional equation

Λ(s)=(4650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46504650    =    2⋅3⋅52⋅312 \cdot 3 \cdot 5^{2} \cdot 31
Sign: 11
Analytic conductor: 37.130437.1304
Root analytic conductor: 6.093476.09347
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4650, ( :1/2), 1)(2,\ 4650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0127720771.012772077
L(12)L(\frac12) ≈\approx 1.0127720771.012772077
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
3 1+T 1 + T
5 1 1
31 1+T 1 + T
good7 1+2T+7T2 1 + 2T + 7T^{2}
11 1−3.26T+11T2 1 - 3.26T + 11T^{2}
13 1+2.73T+13T2 1 + 2.73T + 13T^{2}
17 1−4.93T+17T2 1 - 4.93T + 17T^{2}
19 1−4.93T+19T2 1 - 4.93T + 19T^{2}
23 1−0.521T+23T2 1 - 0.521T + 23T^{2}
29 1−0.521T+29T2 1 - 0.521T + 29T^{2}
37 1−10.8T+37T2 1 - 10.8T + 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+8.82T+43T2 1 + 8.82T + 43T^{2}
47 1+4.93T+47T2 1 + 4.93T + 47T^{2}
53 1+13.3T+53T2 1 + 13.3T + 53T^{2}
59 1−9.34T+59T2 1 - 9.34T + 59T^{2}
61 1+9.75T+61T2 1 + 9.75T + 61T^{2}
67 1−13.5T+67T2 1 - 13.5T + 67T^{2}
71 1−5.56T+71T2 1 - 5.56T + 71T^{2}
73 1−3.04T+73T2 1 - 3.04T + 73T^{2}
79 1+6.41T+79T2 1 + 6.41T + 79T^{2}
83 1−1.36T+83T2 1 - 1.36T + 83T^{2}
89 1+11.8T+89T2 1 + 11.8T + 89T^{2}
97 1−7.26T+97T2 1 - 7.26T + 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

−8.209064497264286700334475362011, −7.59174604127695001783029055667, −6.85302392465612692636374816777, −6.29572385855561449179275244115, −5.53520038971741542371087004209, −4.70707677864288064361336871533, −3.58507823811645053884986134088, −2.91931212057375102680141612124, −1.60293390913198760054675371725, −0.66831876971305387295147282183, 0.66831876971305387295147282183, 1.60293390913198760054675371725, 2.91931212057375102680141612124, 3.58507823811645053884986134088, 4.70707677864288064361336871533, 5.53520038971741542371087004209, 6.29572385855561449179275244115, 6.85302392465612692636374816777, 7.59174604127695001783029055667, 8.209064497264286700334475362011

Graph of the ZZ-function along the critical line