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Properties

Label 2-3332-68.67-c0-0-6
Degree 22
Conductor 33323332
Sign 11
Analytic cond. 1.662881.66288
Root an. cond. 1.289521.28952
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.73·3-s + 4-s + 1.73·6-s − 8-s + 1.99·9-s + 1.73·11-s − 1.73·12-s + 13-s + 16-s + 17-s − 1.99·18-s − 1.73·22-s + 1.73·24-s + 25-s − 26-s − 1.73·27-s − 32-s − 2.99·33-s − 34-s + 1.99·36-s − 1.73·39-s + 1.73·44-s − 1.73·48-s − 50-s − 1.73·51-s + 52-s + ⋯
L(s)  = 1  − 2-s − 1.73·3-s + 4-s + 1.73·6-s − 8-s + 1.99·9-s + 1.73·11-s − 1.73·12-s + 13-s + 16-s + 17-s − 1.99·18-s − 1.73·22-s + 1.73·24-s + 25-s − 26-s − 1.73·27-s − 32-s − 2.99·33-s − 34-s + 1.99·36-s − 1.73·39-s + 1.73·44-s − 1.73·48-s − 50-s − 1.73·51-s + 52-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 11
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(883,⋅)\chi_{3332} (883, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 1)(2,\ 3332,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.57413991580.5741399158
L(12)L(\frac12) ≈\approx 0.57413991580.5741399158
L(1)L(1) 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
7 1 1
17 1−T 1 - T
good3 1+1.73T+T2 1 + 1.73T + T^{2}
5 1−T2 1 - T^{2}
11 1−1.73T+T2 1 - 1.73T + T^{2}
13 1−T+T2 1 - T + T^{2}
19 1−T2 1 - T^{2}
23 1+T2 1 + T^{2}
29 1−T2 1 - T^{2}
31 1+T2 1 + T^{2}
37 1−T2 1 - T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1−T2 1 - T^{2}
53 1+T+T2 1 + T + T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−T2 1 - T^{2}
71 1+1.73T+T2 1 + 1.73T + T^{2}
73 1−T2 1 - T^{2}
79 1−1.73T+T2 1 - 1.73T + T^{2}
83 1−T2 1 - T^{2}
89 1−T+T2 1 - T + T^{2}
97 1−T2 1 - 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

−8.973293562638251319724270053544, −8.056522561671168512024461503494, −7.10350550302749743541646205977, −6.52341598717714413264105181402, −6.07660500680430133180969592635, −5.31004788033584759314288198884, −4.21815871410334199376265175633, −3.29416024314644885509815301305, −1.54204151387104326027237222343, −0.960671531228963139956870116350, 0.960671531228963139956870116350, 1.54204151387104326027237222343, 3.29416024314644885509815301305, 4.21815871410334199376265175633, 5.31004788033584759314288198884, 6.07660500680430133180969592635, 6.52341598717714413264105181402, 7.10350550302749743541646205977, 8.056522561671168512024461503494, 8.973293562638251319724270053544

Graph of the ZZ-function along the critical line