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Properties

Label 2-507-1.1-c3-0-19
Degree 22
Conductor 507507
Sign 11
Analytic cond. 29.913929.9139
Root an. cond. 5.469365.46936
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.36·2-s + 3·3-s + 20.7·4-s + 2.69·5-s − 16.0·6-s + 15.2·7-s − 68.5·8-s + 9·9-s − 14.4·10-s − 66.8·11-s + 62.3·12-s − 81.5·14-s + 8.08·15-s + 201.·16-s + 4.16·17-s − 48.2·18-s + 26.0·19-s + 56.0·20-s + 45.6·21-s + 358.·22-s + 47.3·23-s − 205.·24-s − 117.·25-s + 27·27-s + 315.·28-s + 257.·29-s − 43.3·30-s + ⋯
L(s)  = 1  − 1.89·2-s + 0.577·3-s + 2.59·4-s + 0.241·5-s − 1.09·6-s + 0.820·7-s − 3.03·8-s + 0.333·9-s − 0.457·10-s − 1.83·11-s + 1.49·12-s − 1.55·14-s + 0.139·15-s + 3.14·16-s + 0.0594·17-s − 0.632·18-s + 0.314·19-s + 0.626·20-s + 0.473·21-s + 3.47·22-s + 0.429·23-s − 1.74·24-s − 0.941·25-s + 0.192·27-s + 2.13·28-s + 1.64·29-s − 0.264·30-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 11
Analytic conductor: 29.913929.9139
Root analytic conductor: 5.469365.46936
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 507, ( :3/2), 1)(2,\ 507,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.0369580371.036958037
L(12)L(\frac12) ≈\approx 1.0369580371.036958037
L(52)L(\frac{5}{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 1−3T 1 - 3T
13 1 1
good2 1+5.36T+8T2 1 + 5.36T + 8T^{2}
5 1−2.69T+125T2 1 - 2.69T + 125T^{2}
7 1−15.2T+343T2 1 - 15.2T + 343T^{2}
11 1+66.8T+1.33e3T2 1 + 66.8T + 1.33e3T^{2}
17 1−4.16T+4.91e3T2 1 - 4.16T + 4.91e3T^{2}
19 1−26.0T+6.85e3T2 1 - 26.0T + 6.85e3T^{2}
23 1−47.3T+1.21e4T2 1 - 47.3T + 1.21e4T^{2}
29 1−257.T+2.43e4T2 1 - 257.T + 2.43e4T^{2}
31 1−206.T+2.97e4T2 1 - 206.T + 2.97e4T^{2}
37 1−175.T+5.06e4T2 1 - 175.T + 5.06e4T^{2}
41 1−156.T+6.89e4T2 1 - 156.T + 6.89e4T^{2}
43 1−51.9T+7.95e4T2 1 - 51.9T + 7.95e4T^{2}
47 1+354.T+1.03e5T2 1 + 354.T + 1.03e5T^{2}
53 1+10.4T+1.48e5T2 1 + 10.4T + 1.48e5T^{2}
59 1−445.T+2.05e5T2 1 - 445.T + 2.05e5T^{2}
61 1−119.T+2.26e5T2 1 - 119.T + 2.26e5T^{2}
67 1+22.4T+3.00e5T2 1 + 22.4T + 3.00e5T^{2}
71 1−285.T+3.57e5T2 1 - 285.T + 3.57e5T^{2}
73 1−740.T+3.89e5T2 1 - 740.T + 3.89e5T^{2}
79 1+547.T+4.93e5T2 1 + 547.T + 4.93e5T^{2}
83 1−603.T+5.71e5T2 1 - 603.T + 5.71e5T^{2}
89 1−215.T+7.04e5T2 1 - 215.T + 7.04e5T^{2}
97 1−1.44e3T+9.12e5T2 1 - 1.44e3T + 9.12e5T^{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

−10.18626469744715005230456944119, −9.697453351880264943999908521871, −8.508602949873867661428037617495, −8.057917550880501202535927941494, −7.44918223507081942194450507813, −6.26489176961936937539319986463, −4.96627401109933075391853403715, −2.90055856612341500909020224567, −2.12236369668891410282893892752, −0.808707953307720000271987847266, 0.808707953307720000271987847266, 2.12236369668891410282893892752, 2.90055856612341500909020224567, 4.96627401109933075391853403715, 6.26489176961936937539319986463, 7.44918223507081942194450507813, 8.057917550880501202535927941494, 8.508602949873867661428037617495, 9.697453351880264943999908521871, 10.18626469744715005230456944119

Graph of the ZZ-function along the critical line