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Properties

Label 2-208-13.3-c7-0-14
Degree 22
Conductor 208208
Sign −0.471−0.881i-0.471 - 0.881i
Analytic cond. 64.976064.9760
Root an. cond. 8.060778.06077
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.9 + 27.6i)3-s − 54.4·5-s + (−556. + 963. i)7-s + (585. − 1.01e3i)9-s + (3.56e3 + 6.17e3i)11-s + (7.56e3 + 2.34e3i)13-s + (−867. − 1.50e3i)15-s + (1.02e4 − 1.77e4i)17-s + (1.40e4 − 2.43e4i)19-s − 3.54e4·21-s + (1.69e4 + 2.93e4i)23-s − 7.51e4·25-s + 1.07e5·27-s + (8.77e4 + 1.51e5i)29-s − 1.56e4·31-s + ⋯
L(s)  = 1  + (0.340 + 0.590i)3-s − 0.194·5-s + (−0.613 + 1.06i)7-s + (0.267 − 0.463i)9-s + (0.807 + 1.39i)11-s + (0.955 + 0.296i)13-s + (−0.0663 − 0.114i)15-s + (0.506 − 0.876i)17-s + (0.469 − 0.813i)19-s − 0.835·21-s + (0.290 + 0.503i)23-s − 0.962·25-s + 1.04·27-s + (0.668 + 1.15i)29-s − 0.0945·31-s + ⋯

Functional equation

Λ(s)=(208s/2 ΓC(s) L(s)=((−0.471−0.881i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(208s/2 ΓC(s+7/2) L(s)=((−0.471−0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 208208    =    24⋅132^{4} \cdot 13
Sign: −0.471−0.881i-0.471 - 0.881i
Analytic conductor: 64.976064.9760
Root analytic conductor: 8.060778.06077
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ208(81,⋅)\chi_{208} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 208, ( :7/2), −0.471−0.881i)(2,\ 208,\ (\ :7/2),\ -0.471 - 0.881i)

Particular Values

L(4)L(4) ≈\approx 2.2472687862.247268786
L(12)L(\frac12) ≈\approx 2.2472687862.247268786
L(92)L(\frac{9}{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+(−7.56e3−2.34e3i)T 1 + (-7.56e3 - 2.34e3i)T
good3 1+(−15.9−27.6i)T+(−1.09e3+1.89e3i)T2 1 + (-15.9 - 27.6i)T + (-1.09e3 + 1.89e3i)T^{2}
5 1+54.4T+7.81e4T2 1 + 54.4T + 7.81e4T^{2}
7 1+(556.−963.i)T+(−4.11e5−7.13e5i)T2 1 + (556. - 963. i)T + (-4.11e5 - 7.13e5i)T^{2}
11 1+(−3.56e3−6.17e3i)T+(−9.74e6+1.68e7i)T2 1 + (-3.56e3 - 6.17e3i)T + (-9.74e6 + 1.68e7i)T^{2}
17 1+(−1.02e4+1.77e4i)T+(−2.05e8−3.55e8i)T2 1 + (-1.02e4 + 1.77e4i)T + (-2.05e8 - 3.55e8i)T^{2}
19 1+(−1.40e4+2.43e4i)T+(−4.46e8−7.74e8i)T2 1 + (-1.40e4 + 2.43e4i)T + (-4.46e8 - 7.74e8i)T^{2}
23 1+(−1.69e4−2.93e4i)T+(−1.70e9+2.94e9i)T2 1 + (-1.69e4 - 2.93e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(−8.77e4−1.51e5i)T+(−8.62e9+1.49e10i)T2 1 + (-8.77e4 - 1.51e5i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+1.56e4T+2.75e10T2 1 + 1.56e4T + 2.75e10T^{2}
37 1+(998.+1.72e3i)T+(−4.74e10+8.22e10i)T2 1 + (998. + 1.72e3i)T + (-4.74e10 + 8.22e10i)T^{2}
41 1+(1.11e5+1.93e5i)T+(−9.73e10+1.68e11i)T2 1 + (1.11e5 + 1.93e5i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1+(2.68e5−4.65e5i)T+(−1.35e11−2.35e11i)T2 1 + (2.68e5 - 4.65e5i)T + (-1.35e11 - 2.35e11i)T^{2}
47 1+5.42e5T+5.06e11T2 1 + 5.42e5T + 5.06e11T^{2}
53 1−1.85e6T+1.17e12T2 1 - 1.85e6T + 1.17e12T^{2}
59 1+(6.65e5−1.15e6i)T+(−1.24e12−2.15e12i)T2 1 + (6.65e5 - 1.15e6i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(1.43e6−2.48e6i)T+(−1.57e12−2.72e12i)T2 1 + (1.43e6 - 2.48e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(1.41e6+2.45e6i)T+(−3.03e12+5.24e12i)T2 1 + (1.41e6 + 2.45e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+(8.00e5−1.38e6i)T+(−4.54e12−7.87e12i)T2 1 + (8.00e5 - 1.38e6i)T + (-4.54e12 - 7.87e12i)T^{2}
73 1−1.39e6T+1.10e13T2 1 - 1.39e6T + 1.10e13T^{2}
79 1−2.33e6T+1.92e13T2 1 - 2.33e6T + 1.92e13T^{2}
83 1+2.37e6T+2.71e13T2 1 + 2.37e6T + 2.71e13T^{2}
89 1+(3.52e6+6.10e6i)T+(−2.21e13+3.83e13i)T2 1 + (3.52e6 + 6.10e6i)T + (-2.21e13 + 3.83e13i)T^{2}
97 1+(4.25e6−7.37e6i)T+(−4.03e13−6.99e13i)T2 1 + (4.25e6 - 7.37e6i)T + (-4.03e13 - 6.99e13i)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

−11.69287593306671377084663954763, −10.17238598708552694689047551257, −9.297142821605990467825386387463, −8.934594298719970194408615138961, −7.30040065752265671448300821105, −6.38146268155069905173905980654, −5.02680450577054515499015346934, −3.87322850338985657369753701000, −2.85314038969573327238300437575, −1.31043157954465461931493570673, 0.58235615849731165194060371079, 1.51098374215917789527966705382, 3.29341464611578854888815881047, 4.01862030606725555017088645683, 5.86511987164250954573639045910, 6.70556248336846947539670702942, 7.901621637259656658246224133943, 8.482957551806704341292879302050, 9.955320639354248175210098142638, 10.73115309618100412777918850967

Graph of the ZZ-function along the critical line