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Properties

Label 2-416-52.3-c0-0-0
Degree 22
Conductor 416416
Sign 0.494−0.869i0.494 - 0.869i
Analytic cond. 0.2076110.207611
Root an. cond. 0.4556430.455643
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)3-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + 13-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − 0.999·21-s + (0.866 − 0.5i)23-s − 25-s − i·27-s + (0.5 + 0.866i)29-s + (0.499 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (−0.866 + 0.5i)39-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + 13-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − 0.999·21-s + (0.866 − 0.5i)23-s − 25-s − i·27-s + (0.5 + 0.866i)29-s + (0.499 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (−0.866 + 0.5i)39-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.494−0.869i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s) L(s)=((0.494−0.869i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.494−0.869i0.494 - 0.869i
Analytic conductor: 0.2076110.207611
Root analytic conductor: 0.4556430.455643
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ416(159,⋅)\chi_{416} (159, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :0), 0.494−0.869i)(2,\ 416,\ (\ :0),\ 0.494 - 0.869i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.67605012870.6760501287
L(12)L(\frac12) ≈\approx 0.67605012870.6760501287
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 1
13 1−T 1 - T
good3 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
5 1+T2 1 + T^{2}
7 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
17 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
19 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
23 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
29 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
31 1−T2 1 - T^{2}
37 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
41 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-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+2iT−T2 1 + 2iT - T^{2}
53 1+T2 1 + T^{2}
59 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
61 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
71 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
73 1+T2 1 + T^{2}
79 1−2iT−T2 1 - 2iT - T^{2}
83 1−T2 1 - T^{2}
89 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
97 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-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

−11.45407323113429728214257958095, −10.71453289582306159245829494502, −10.12554656228783781527044351141, −8.726316963686132906655453740779, −8.082139972157866462866888207237, −6.78913577969456354190091977661, −5.50067776687963736586586474262, −5.12924886940871015484160370661, −3.80493148432603117085106764238, −2.01481993602685396123698186653, 1.16817482024152375815029543823, 3.09689828821238581465743595102, 4.67881471752009105687629546664, 5.55519337493800148989496338698, 6.53658872710570387340612681246, 7.52366621259775273015933246989, 8.362350221878164128253945514060, 9.525949676586403847380620967776, 10.71647469203674048902874289702, 11.46235892079518807258944203133

Graph of the ZZ-function along the critical line