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Properties

Label 2-675-1.1-c3-0-51
Degree 22
Conductor 675675
Sign 11
Analytic cond. 39.826239.8262
Root an. cond. 6.310806.31080
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  + 4.72·2-s + 14.3·4-s + 17.6·7-s + 29.9·8-s + 34.2·11-s + 53.8·13-s + 83.3·14-s + 26.9·16-s − 74.7·17-s − 89.5·19-s + 161.·22-s + 176.·23-s + 254.·26-s + 252.·28-s + 194.·29-s + 107.·31-s − 112.·32-s − 353.·34-s − 430.·37-s − 423.·38-s − 108.·41-s + 409.·43-s + 490.·44-s + 832.·46-s + 409.·47-s − 32.3·49-s + 772.·52-s + ⋯
L(s)  = 1  + 1.67·2-s + 1.79·4-s + 0.951·7-s + 1.32·8-s + 0.938·11-s + 1.14·13-s + 1.59·14-s + 0.420·16-s − 1.06·17-s − 1.08·19-s + 1.56·22-s + 1.59·23-s + 1.92·26-s + 1.70·28-s + 1.24·29-s + 0.625·31-s − 0.621·32-s − 1.78·34-s − 1.91·37-s − 1.80·38-s − 0.414·41-s + 1.45·43-s + 1.68·44-s + 2.66·46-s + 1.26·47-s − 0.0942·49-s + 2.06·52-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 39.826239.8262
Root analytic conductor: 6.310806.31080
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 675, ( :3/2), 1)(2,\ 675,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 6.5950648936.595064893
L(12)L(\frac12) ≈\approx 6.5950648936.595064893
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 1
5 1 1
good2 1−4.72T+8T2 1 - 4.72T + 8T^{2}
7 1−17.6T+343T2 1 - 17.6T + 343T^{2}
11 1−34.2T+1.33e3T2 1 - 34.2T + 1.33e3T^{2}
13 1−53.8T+2.19e3T2 1 - 53.8T + 2.19e3T^{2}
17 1+74.7T+4.91e3T2 1 + 74.7T + 4.91e3T^{2}
19 1+89.5T+6.85e3T2 1 + 89.5T + 6.85e3T^{2}
23 1−176.T+1.21e4T2 1 - 176.T + 1.21e4T^{2}
29 1−194.T+2.43e4T2 1 - 194.T + 2.43e4T^{2}
31 1−107.T+2.97e4T2 1 - 107.T + 2.97e4T^{2}
37 1+430.T+5.06e4T2 1 + 430.T + 5.06e4T^{2}
41 1+108.T+6.89e4T2 1 + 108.T + 6.89e4T^{2}
43 1−409.T+7.95e4T2 1 - 409.T + 7.95e4T^{2}
47 1−409.T+1.03e5T2 1 - 409.T + 1.03e5T^{2}
53 1+24.7T+1.48e5T2 1 + 24.7T + 1.48e5T^{2}
59 1+295.T+2.05e5T2 1 + 295.T + 2.05e5T^{2}
61 1−305.T+2.26e5T2 1 - 305.T + 2.26e5T^{2}
67 1−915.T+3.00e5T2 1 - 915.T + 3.00e5T^{2}
71 1−228.T+3.57e5T2 1 - 228.T + 3.57e5T^{2}
73 1+158.T+3.89e5T2 1 + 158.T + 3.89e5T^{2}
79 1+319.T+4.93e5T2 1 + 319.T + 4.93e5T^{2}
83 1+936.T+5.71e5T2 1 + 936.T + 5.71e5T^{2}
89 1−920.T+7.04e5T2 1 - 920.T + 7.04e5T^{2}
97 1+914.T+9.12e5T2 1 + 914.T + 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

−10.74876727449536854510236490957, −9.000357740098645967644555965293, −8.409392055075721851563552251319, −6.90111400305564020519232836489, −6.46258074171494389415915486706, −5.35175842239200626864032898204, −4.48961372882132526124054158582, −3.83307642005460006128568726533, −2.57388877336550929259719498313, −1.35769056484624897502867917158, 1.35769056484624897502867917158, 2.57388877336550929259719498313, 3.83307642005460006128568726533, 4.48961372882132526124054158582, 5.35175842239200626864032898204, 6.46258074171494389415915486706, 6.90111400305564020519232836489, 8.409392055075721851563552251319, 9.000357740098645967644555965293, 10.74876727449536854510236490957

Graph of the ZZ-function along the critical line