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Properties

Label 2-702-1.1-c1-0-5
Degree 22
Conductor 702702
Sign 11
Analytic cond. 5.605495.60549
Root an. cond. 2.367592.36759
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s + 3·11-s + 13-s − 14-s + 16-s + 6·17-s + 2·19-s + 3·22-s + 3·23-s − 5·25-s + 26-s − 28-s + 3·29-s + 5·31-s + 32-s + 6·34-s − 7·37-s + 2·38-s − 6·41-s − 43-s + 3·44-s + 3·46-s − 6·49-s − 5·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s + 0.639·22-s + 0.625·23-s − 25-s + 0.196·26-s − 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s + 1.02·34-s − 1.15·37-s + 0.324·38-s − 0.937·41-s − 0.152·43-s + 0.452·44-s + 0.442·46-s − 6/7·49-s − 0.707·50-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: 11
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), 1)(2,\ 702,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4387240232.438724023
L(12)L(\frac12) ≈\approx 2.4387240232.438724023
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 1
13 1−T 1 - T
good5 1+pT2 1 + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1+3T+pT2 1 + 3 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1+9T+pT2 1 + 9 T + p T^{2}
89 1−3T+pT2 1 - 3 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p T^{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

−10.40177030669352365127911103866, −9.756591550306942426380153961599, −8.714102464355361131347704225310, −7.68464846057382852123303689608, −6.75704327285852474476313882161, −5.94796173241577251258054909189, −5.00762274654052120160679380267, −3.82098806276952507517625807862, −3.04361032800015499774600610053, −1.39366852032118049822765695687, 1.39366852032118049822765695687, 3.04361032800015499774600610053, 3.82098806276952507517625807862, 5.00762274654052120160679380267, 5.94796173241577251258054909189, 6.75704327285852474476313882161, 7.68464846057382852123303689608, 8.714102464355361131347704225310, 9.756591550306942426380153961599, 10.40177030669352365127911103866

Graph of the ZZ-function along the critical line