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Properties

Label 2-3328-1.1-c1-0-12
Degree 22
Conductor 33283328
Sign 11
Analytic cond. 26.574226.5742
Root an. cond. 5.155015.15501
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  + 2·3-s − 3.46·5-s − 4.73·7-s + 9-s + 1.26·11-s − 13-s − 6.92·15-s − 1.46·17-s − 2.73·19-s − 9.46·21-s + 4·23-s + 6.99·25-s − 4·27-s − 2·29-s + 3.26·31-s + 2.53·33-s + 16.3·35-s + 4.92·37-s − 2·39-s + 4.92·41-s + 7.46·43-s − 3.46·45-s − 3.26·47-s + 15.3·49-s − 2.92·51-s + 10.9·53-s − 4.39·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.54·5-s − 1.78·7-s + 0.333·9-s + 0.382·11-s − 0.277·13-s − 1.78·15-s − 0.355·17-s − 0.626·19-s − 2.06·21-s + 0.834·23-s + 1.39·25-s − 0.769·27-s − 0.371·29-s + 0.586·31-s + 0.441·33-s + 2.77·35-s + 0.810·37-s − 0.320·39-s + 0.769·41-s + 1.13·43-s − 0.516·45-s − 0.476·47-s + 2.19·49-s − 0.410·51-s + 1.50·53-s − 0.592·55-s + ⋯

Functional equation

Λ(s)=(3328s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3328s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33283328    =    28⋅132^{8} \cdot 13
Sign: 11
Analytic conductor: 26.574226.5742
Root analytic conductor: 5.155015.15501
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3328, ( :1/2), 1)(2,\ 3328,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1878450931.187845093
L(12)L(\frac12) ≈\approx 1.1878450931.187845093
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
13 1+T 1 + T
good3 1−2T+3T2 1 - 2T + 3T^{2}
5 1+3.46T+5T2 1 + 3.46T + 5T^{2}
7 1+4.73T+7T2 1 + 4.73T + 7T^{2}
11 1−1.26T+11T2 1 - 1.26T + 11T^{2}
17 1+1.46T+17T2 1 + 1.46T + 17T^{2}
19 1+2.73T+19T2 1 + 2.73T + 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−3.26T+31T2 1 - 3.26T + 31T^{2}
37 1−4.92T+37T2 1 - 4.92T + 37T^{2}
41 1−4.92T+41T2 1 - 4.92T + 41T^{2}
43 1−7.46T+43T2 1 - 7.46T + 43T^{2}
47 1+3.26T+47T2 1 + 3.26T + 47T^{2}
53 1−10.9T+53T2 1 - 10.9T + 53T^{2}
59 1+0.196T+59T2 1 + 0.196T + 59T^{2}
61 1−10.9T+61T2 1 - 10.9T + 61T^{2}
67 1+2.73T+67T2 1 + 2.73T + 67T^{2}
71 1−2.19T+71T2 1 - 2.19T + 71T^{2}
73 1−0.535T+73T2 1 - 0.535T + 73T^{2}
79 1−1.46T+79T2 1 - 1.46T + 79T^{2}
83 1−6.73T+83T2 1 - 6.73T + 83T^{2}
89 1+17.3T+89T2 1 + 17.3T + 89T^{2}
97 1+14.3T+97T2 1 + 14.3T + 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

−8.622076288579392155964461226193, −7.975975262870032889106454267237, −7.17236414352687234796768068874, −6.71240091680589272447303787065, −5.70495272670373130184862222283, −4.26181703856032238634869666773, −3.87633909156815732243980757314, −3.06324705379444144622456790442, −2.50955288530124832694397755115, −0.58137463746360388236987393310, 0.58137463746360388236987393310, 2.50955288530124832694397755115, 3.06324705379444144622456790442, 3.87633909156815732243980757314, 4.26181703856032238634869666773, 5.70495272670373130184862222283, 6.71240091680589272447303787065, 7.17236414352687234796768068874, 7.975975262870032889106454267237, 8.622076288579392155964461226193

Graph of the ZZ-function along the critical line