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Properties

Label 2-28e2-112.107-c0-0-0
Degree 22
Conductor 784784
Sign 0.496−0.868i0.496 - 0.868i
Analytic cond. 0.3912660.391266
Root an. cond. 0.6255130.625513
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (0.866 + 0.5i)9-s + (−1.36 − 0.366i)11-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + (−0.999 − i)22-s + (1 − 1.73i)23-s + (−0.866 + 0.5i)25-s + (1 − i)29-s + (−0.866 + 0.499i)32-s + 0.999i·36-s + (−1.36 + 0.366i)37-s + (−1 + i)43-s + (−0.366 − 1.36i)44-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (0.866 + 0.5i)9-s + (−1.36 − 0.366i)11-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + (−0.999 − i)22-s + (1 − 1.73i)23-s + (−0.866 + 0.5i)25-s + (1 − i)29-s + (−0.866 + 0.499i)32-s + 0.999i·36-s + (−1.36 + 0.366i)37-s + (−1 + i)43-s + (−0.366 − 1.36i)44-s + ⋯

Functional equation

Λ(s)=(784s/2 ΓC(s) L(s)=((0.496−0.868i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(784s/2 ΓC(s) L(s)=((0.496−0.868i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 784784    =    24⋅722^{4} \cdot 7^{2}
Sign: 0.496−0.868i0.496 - 0.868i
Analytic conductor: 0.3912660.391266
Root analytic conductor: 0.6255130.625513
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ784(667,⋅)\chi_{784} (667, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 784, ( :0), 0.496−0.868i)(2,\ 784,\ (\ :0),\ 0.496 - 0.868i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5413340901.541334090
L(12)L(\frac12) ≈\approx 1.5413340901.541334090
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
7 1 1
good3 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
5 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
11 1+(1.36+0.366i)T+(0.866+0.5i)T2 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2}
13 1−iT2 1 - iT^{2}
17 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
19 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
23 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
29 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(1.36−0.366i)T+(0.866−0.5i)T2 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2}
41 1−T2 1 - T^{2}
43 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(−0.366+1.36i)T+(−0.866−0.5i)T2 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2}
59 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
61 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
67 1+(−0.366+1.36i)T+(−0.866−0.5i)T2 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2}
71 1+T2 1 + T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(1.73+i)T+(0.5+0.866i)T2 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2}
83 1+iT2 1 + iT^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)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

−10.66902795580643181635394708897, −10.02910475939848803816805734859, −8.564371068402713343074600174251, −7.962348204137922259291116922026, −7.09698757218468603499530286888, −6.25110118175128429337876503529, −5.11964682627567840236611993304, −4.58336630805791750171568083605, −3.27536228670837657177471745648, −2.21345868600080706104598064328, 1.57334716477303286272011606463, 2.89195887764347738093865720242, 3.88932592050006074511414690820, 4.96408329511130940346901685193, 5.63650562291649736988102925566, 6.90661095712952450031109156460, 7.44555998490019468433815833198, 8.816207712265261560324684130014, 9.936162844383575306814909795876, 10.30681331664546908427426373300

Graph of the ZZ-function along the critical line