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Properties

Label 2-296-1.1-c1-0-1
Degree 22
Conductor 296296
Sign 11
Analytic cond. 2.363572.36357
Root an. cond. 1.537391.53739
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  − 1.93·3-s − 2.93·5-s + 4.68·7-s + 0.745·9-s + 0.762·11-s + 1.76·13-s + 5.68·15-s + 3.36·17-s + 7.36·19-s − 9.06·21-s + 3.25·23-s + 3.61·25-s + 4.36·27-s − 3.25·29-s + 3.06·31-s − 1.47·33-s − 13.7·35-s − 37-s − 3.41·39-s − 7.42·41-s − 12.2·43-s − 2.18·45-s + 0.302·47-s + 14.9·49-s − 6.50·51-s + 5.53·53-s − 2.23·55-s + ⋯
L(s)  = 1  − 1.11·3-s − 1.31·5-s + 1.76·7-s + 0.248·9-s + 0.229·11-s + 0.488·13-s + 1.46·15-s + 0.815·17-s + 1.68·19-s − 1.97·21-s + 0.678·23-s + 0.723·25-s + 0.839·27-s − 0.604·29-s + 0.550·31-s − 0.256·33-s − 2.32·35-s − 0.164·37-s − 0.546·39-s − 1.15·41-s − 1.86·43-s − 0.326·45-s + 0.0440·47-s + 2.13·49-s − 0.911·51-s + 0.760·53-s − 0.301·55-s + ⋯

Functional equation

Λ(s)=(296s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(296s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 296296    =    23⋅372^{3} \cdot 37
Sign: 11
Analytic conductor: 2.363572.36357
Root analytic conductor: 1.537391.53739
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 296, ( :1/2), 1)(2,\ 296,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.88683952930.8868395293
L(12)L(\frac12) ≈\approx 0.88683952930.8868395293
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
37 1+T 1 + T
good3 1+1.93T+3T2 1 + 1.93T + 3T^{2}
5 1+2.93T+5T2 1 + 2.93T + 5T^{2}
7 1−4.68T+7T2 1 - 4.68T + 7T^{2}
11 1−0.762T+11T2 1 - 0.762T + 11T^{2}
13 1−1.76T+13T2 1 - 1.76T + 13T^{2}
17 1−3.36T+17T2 1 - 3.36T + 17T^{2}
19 1−7.36T+19T2 1 - 7.36T + 19T^{2}
23 1−3.25T+23T2 1 - 3.25T + 23T^{2}
29 1+3.25T+29T2 1 + 3.25T + 29T^{2}
31 1−3.06T+31T2 1 - 3.06T + 31T^{2}
41 1+7.42T+41T2 1 + 7.42T + 41T^{2}
43 1+12.2T+43T2 1 + 12.2T + 43T^{2}
47 1−0.302T+47T2 1 - 0.302T + 47T^{2}
53 1−5.53T+53T2 1 - 5.53T + 53T^{2}
59 1−10.2T+59T2 1 - 10.2T + 59T^{2}
61 1−12.2T+61T2 1 - 12.2T + 61T^{2}
67 1+13.1T+67T2 1 + 13.1T + 67T^{2}
71 1−0.173T+71T2 1 - 0.173T + 71T^{2}
73 1+1.23T+73T2 1 + 1.23T + 73T^{2}
79 1−4.61T+79T2 1 - 4.61T + 79T^{2}
83 1−3.53T+83T2 1 - 3.53T + 83T^{2}
89 1−15.7T+89T2 1 - 15.7T + 89T^{2}
97 1+16.1T+97T2 1 + 16.1T + 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

−11.65184239375590915359083015172, −11.29121850641396720850685794947, −10.24559456423596935575688897165, −8.627342280285411058871019505602, −7.88637576847425579767411056500, −7.01388337253448340997932032971, −5.46159198185251203645560997923, −4.86530737922999155740176177980, −3.56361983012300239336936775696, −1.12037709954869759125968838726, 1.12037709954869759125968838726, 3.56361983012300239336936775696, 4.86530737922999155740176177980, 5.46159198185251203645560997923, 7.01388337253448340997932032971, 7.88637576847425579767411056500, 8.627342280285411058871019505602, 10.24559456423596935575688897165, 11.29121850641396720850685794947, 11.65184239375590915359083015172

Graph of the ZZ-function along the critical line