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Properties

Label 2-2268-252.167-c0-0-12
Degree 22
Conductor 22682268
Sign −0.0871+0.996i-0.0871 + 0.996i
Analytic cond. 1.131871.13187
Root an. cond. 1.063891.06389
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.258 − 0.448i)11-s + (−0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (−0.366 + 0.366i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (−1.22 + 0.707i)29-s + (−0.965 − 0.258i)32-s + 1.73·37-s + (0.866 − 0.5i)43-s + (0.448 + 0.258i)44-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.258 − 0.448i)11-s + (−0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (−0.366 + 0.366i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (−1.22 + 0.707i)29-s + (−0.965 − 0.258i)32-s + 1.73·37-s + (0.866 − 0.5i)43-s + (0.448 + 0.258i)44-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=((−0.0871+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s) L(s)=((−0.0871+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: −0.0871+0.996i-0.0871 + 0.996i
Analytic conductor: 1.131871.13187
Root analytic conductor: 1.063891.06389
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2268(1511,⋅)\chi_{2268} (1511, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2268, ( :0), −0.0871+0.996i)(2,\ 2268,\ (\ :0),\ -0.0871 + 0.996i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0343840521.034384052
L(12)L(\frac12) ≈\approx 1.0343840521.034384052
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+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
3 1 1
7 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
good5 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
11 1+(0.258+0.448i)T+(−0.5+0.866i)T2 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2}
13 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+T2 1 + T^{2}
19 1+T2 1 + T^{2}
23 1+(−0.707+1.22i)T+(−0.5−0.866i)T2 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}
29 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1−1.73T+T2 1 - 1.73T + T^{2}
41 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
43 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+1.93iT−T2 1 + 1.93iT - T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
71 1−1.93T+T2 1 - 1.93T + T^{2}
73 1−T2 1 - T^{2}
79 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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

−9.050716138574731087306870687636, −8.375309797721427020229224676898, −7.69248255533790781423757528139, −6.90493596845983963406939071769, −5.57054250554653271857398773354, −4.84571140127425227615399944438, −4.03588064868000120881986775735, −3.10619868684909420804412692441, −2.07380927584292872184408146141, −0.929682143265937135147022561357, 1.32470931576504892362420045390, 2.61033707195885073236433122931, 4.10224573903107779714782927856, 4.76889316671184652388922832133, 5.62258129401992267151865755718, 6.19195611297652923253820468572, 7.43948542258157157417132923610, 7.64065635388920878455822876195, 8.579897997055495645616367747640, 9.253076690162030301501300776450

Graph of the ZZ-function along the critical line