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Properties

Label 2-4002-1.1-c1-0-11
Degree 22
Conductor 40024002
Sign 11
Analytic cond. 31.956131.9561
Root an. cond. 5.652975.65297
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-s + 3-s + 4-s − 4.09·5-s + 6-s − 2.85·7-s + 8-s + 9-s − 4.09·10-s + 0.493·11-s + 12-s − 5.03·13-s − 2.85·14-s − 4.09·15-s + 16-s − 0.585·17-s + 18-s + 0.996·19-s − 4.09·20-s − 2.85·21-s + 0.493·22-s + 23-s + 24-s + 11.7·25-s − 5.03·26-s + 27-s − 2.85·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.82·5-s + 0.408·6-s − 1.07·7-s + 0.353·8-s + 0.333·9-s − 1.29·10-s + 0.148·11-s + 0.288·12-s − 1.39·13-s − 0.762·14-s − 1.05·15-s + 0.250·16-s − 0.141·17-s + 0.235·18-s + 0.228·19-s − 0.914·20-s − 0.622·21-s + 0.105·22-s + 0.208·23-s + 0.204·24-s + 2.34·25-s − 0.987·26-s + 0.192·27-s − 0.539·28-s + ⋯

Functional equation

Λ(s)=(4002s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4002s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40024002    =    2⋅3⋅23⋅292 \cdot 3 \cdot 23 \cdot 29
Sign: 11
Analytic conductor: 31.956131.9561
Root analytic conductor: 5.652975.65297
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4002, ( :1/2), 1)(2,\ 4002,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8330682791.833068279
L(12)L(\frac12) ≈\approx 1.8330682791.833068279
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−T 1 - T
23 1−T 1 - T
29 1+T 1 + T
good5 1+4.09T+5T2 1 + 4.09T + 5T^{2}
7 1+2.85T+7T2 1 + 2.85T + 7T^{2}
11 1−0.493T+11T2 1 - 0.493T + 11T^{2}
13 1+5.03T+13T2 1 + 5.03T + 13T^{2}
17 1+0.585T+17T2 1 + 0.585T + 17T^{2}
19 1−0.996T+19T2 1 - 0.996T + 19T^{2}
31 1−4.90T+31T2 1 - 4.90T + 31T^{2}
37 1+1.11T+37T2 1 + 1.11T + 37T^{2}
41 1−7.29T+41T2 1 - 7.29T + 41T^{2}
43 1−6.49T+43T2 1 - 6.49T + 43T^{2}
47 1−4.69T+47T2 1 - 4.69T + 47T^{2}
53 1−7.44T+53T2 1 - 7.44T + 53T^{2}
59 1−1.14T+59T2 1 - 1.14T + 59T^{2}
61 1+6.67T+61T2 1 + 6.67T + 61T^{2}
67 1−16.1T+67T2 1 - 16.1T + 67T^{2}
71 1−6.89T+71T2 1 - 6.89T + 71T^{2}
73 1−1.12T+73T2 1 - 1.12T + 73T^{2}
79 1+10.1T+79T2 1 + 10.1T + 79T^{2}
83 1+0.118T+83T2 1 + 0.118T + 83T^{2}
89 1+12.2T+89T2 1 + 12.2T + 89T^{2}
97 1−1.37T+97T2 1 - 1.37T + 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

−8.271007537483241312296107881368, −7.50850245948629598537655109300, −7.16469334894771380670589032414, −6.41376255720479517770213056365, −5.25855755585126495031644283423, −4.37663894612099892163713793325, −3.89938688455993292908169592663, −3.07649981055823145114303463744, −2.49966754385172427101149416578, −0.64937566553249103465873210922, 0.64937566553249103465873210922, 2.49966754385172427101149416578, 3.07649981055823145114303463744, 3.89938688455993292908169592663, 4.37663894612099892163713793325, 5.25855755585126495031644283423, 6.41376255720479517770213056365, 7.16469334894771380670589032414, 7.50850245948629598537655109300, 8.271007537483241312296107881368

Graph of the ZZ-function along the critical line