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Properties

Label 2-14e2-1.1-c1-0-0
Degree 22
Conductor 196196
Sign 11
Analytic cond. 1.565061.56506
Root an. cond. 1.251021.25102
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  − 2.82·3-s + 1.41·5-s + 5.00·9-s + 4·11-s + 4.24·13-s − 4.00·15-s + 1.41·17-s + 2.82·19-s − 4·23-s − 2.99·25-s − 5.65·27-s + 8·29-s − 11.3·33-s − 8·37-s − 12·39-s − 7.07·41-s − 4·43-s + 7.07·45-s + 5.65·47-s − 4.00·51-s + 10·53-s + 5.65·55-s − 8.00·57-s + 14.1·59-s − 7.07·61-s + 6·65-s + 11.3·69-s + ⋯
L(s)  = 1  − 1.63·3-s + 0.632·5-s + 1.66·9-s + 1.20·11-s + 1.17·13-s − 1.03·15-s + 0.342·17-s + 0.648·19-s − 0.834·23-s − 0.599·25-s − 1.08·27-s + 1.48·29-s − 1.96·33-s − 1.31·37-s − 1.92·39-s − 1.10·41-s − 0.609·43-s + 1.05·45-s + 0.825·47-s − 0.560·51-s + 1.37·53-s + 0.762·55-s − 1.05·57-s + 1.84·59-s − 0.905·61-s + 0.744·65-s + 1.36·69-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 1.565061.56506
Root analytic conductor: 1.251021.25102
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 196, ( :1/2), 1)(2,\ 196,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.87236412700.8723641270
L(12)L(\frac12) ≈\approx 0.87236412700.8723641270
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 1
7 1 1
good3 1+2.82T+3T2 1 + 2.82T + 3T^{2}
5 1−1.41T+5T2 1 - 1.41T + 5T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1−4.24T+13T2 1 - 4.24T + 13T^{2}
17 1−1.41T+17T2 1 - 1.41T + 17T^{2}
19 1−2.82T+19T2 1 - 2.82T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−8T+29T2 1 - 8T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+8T+37T2 1 + 8T + 37T^{2}
41 1+7.07T+41T2 1 + 7.07T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1−5.65T+47T2 1 - 5.65T + 47T^{2}
53 1−10T+53T2 1 - 10T + 53T^{2}
59 1−14.1T+59T2 1 - 14.1T + 59T^{2}
61 1+7.07T+61T2 1 + 7.07T + 61T^{2}
67 1+67T2 1 + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+7.07T+73T2 1 + 7.07T + 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+14.1T+83T2 1 + 14.1T + 83T^{2}
89 1−7.07T+89T2 1 - 7.07T + 89T^{2}
97 1+1.41T+97T2 1 + 1.41T + 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

−12.01801209257579565799880886231, −11.78118085089026838605485024886, −10.58045995448057079474808428178, −9.877393439712082231162545028013, −8.587224786167342026200073187279, −6.89650934972302960219554185576, −6.14539973457519346514706405452, −5.33135340261650426661592219336, −3.91584139815081451345714498381, −1.32286566317686927886391412472, 1.32286566317686927886391412472, 3.91584139815081451345714498381, 5.33135340261650426661592219336, 6.14539973457519346514706405452, 6.89650934972302960219554185576, 8.587224786167342026200073187279, 9.877393439712082231162545028013, 10.58045995448057079474808428178, 11.78118085089026838605485024886, 12.01801209257579565799880886231

Graph of the ZZ-function along the critical line