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Properties

Label 2-1332-148.147-c0-0-0
Degree 22
Conductor 13321332
Sign −i-i
Analytic cond. 0.6647540.664754
Root an. cond. 0.8153240.815324
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + 1.41i·5-s + 2i·7-s + (0.707 − 0.707i)8-s + (1.00 − 1.00i)10-s + (1.41 − 1.41i)14-s − 1.00·16-s − 1.41i·17-s − 1.41·20-s − 1.41·23-s − 1.00·25-s − 2.00·28-s + 1.41i·29-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + 1.41i·5-s + 2i·7-s + (0.707 − 0.707i)8-s + (1.00 − 1.00i)10-s + (1.41 − 1.41i)14-s − 1.00·16-s − 1.41i·17-s − 1.41·20-s − 1.41·23-s − 1.00·25-s − 2.00·28-s + 1.41i·29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

Λ(s)=(1332s/2 ΓC(s) L(s)=(−i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1332s/2 ΓC(s) L(s)=(−i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13321332    =    22⋅32⋅372^{2} \cdot 3^{2} \cdot 37
Sign: −i-i
Analytic conductor: 0.6647540.664754
Root analytic conductor: 0.8153240.815324
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1332(739,⋅)\chi_{1332} (739, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1332, ( :0), −i)(2,\ 1332,\ (\ :0),\ -i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.66024245080.6602424508
L(12)L(\frac12) ≈\approx 0.66024245080.6602424508
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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
3 1 1
37 1−T 1 - T
good5 1−1.41iT−T2 1 - 1.41iT - T^{2}
7 1−2iT−T2 1 - 2iT - T^{2}
11 1−T2 1 - T^{2}
13 1−T2 1 - T^{2}
17 1+1.41iT−T2 1 + 1.41iT - T^{2}
19 1+T2 1 + T^{2}
23 1+1.41T+T2 1 + 1.41T + T^{2}
29 1−1.41iT−T2 1 - 1.41iT - T^{2}
31 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+T2 1 + T^{2}
59 1−1.41T+T2 1 - 1.41T + T^{2}
61 1−T2 1 - T^{2}
67 1+2iT−T2 1 + 2iT - T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1+T2 1 + T^{2}
83 1−T2 1 - T^{2}
89 1−1.41iT−T2 1 - 1.41iT - 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

−9.892056536591717787071047078648, −9.371386255470186008935881934893, −8.571377289541421886647726263158, −7.74331922563593644290663017934, −6.85386042430630659094192996300, −6.05324551244972753635695859349, −4.97076956644728292778138433325, −3.46361225721334591699090925779, −2.70425552170417996162447736291, −2.10248082468340332890912845936, 0.71403658330381082758101991000, 1.72900014117092613444099523696, 4.10672817019103064091094510561, 4.35000340292243973979457524416, 5.60515086101696016820708607005, 6.39576061096915889048739935070, 7.40190470981591793843066613870, 8.081127869578809371764089097926, 8.533047380232175134344017996767, 9.758375538032464746146122612807

Graph of the ZZ-function along the critical line