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Properties

Label 2-464-116.19-c1-0-12
Degree 22
Conductor 464464
Sign −0.485+0.874i-0.485 + 0.874i
Analytic cond. 3.705053.70505
Root an. cond. 1.924851.92485
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.432 + 0.151i)3-s + (−1.40 − 0.320i)5-s + (−0.759 − 1.57i)7-s + (−2.18 − 1.73i)9-s + (0.426 − 3.78i)11-s + (−3.71 + 2.96i)13-s + (−0.558 − 0.350i)15-s + (1.08 − 1.08i)17-s + (−1.57 − 4.48i)19-s + (−0.0898 − 0.797i)21-s + (−0.0809 + 0.0184i)23-s + (−2.64 − 1.27i)25-s + (−1.41 − 2.24i)27-s + (3.03 + 4.44i)29-s + (8.91 − 5.60i)31-s + ⋯
L(s)  = 1  + (0.249 + 0.0873i)3-s + (−0.627 − 0.143i)5-s + (−0.287 − 0.596i)7-s + (−0.727 − 0.579i)9-s + (0.128 − 1.14i)11-s + (−1.03 + 0.821i)13-s + (−0.144 − 0.0905i)15-s + (0.263 − 0.263i)17-s + (−0.360 − 1.03i)19-s + (−0.0196 − 0.174i)21-s + (−0.0168 + 0.00385i)23-s + (−0.528 − 0.254i)25-s + (−0.271 − 0.432i)27-s + (0.563 + 0.826i)29-s + (1.60 − 1.00i)31-s + ⋯

Functional equation

Λ(s)=(464s/2 ΓC(s) L(s)=((−0.485+0.874i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(464s/2 ΓC(s+1/2) L(s)=((−0.485+0.874i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 464464    =    24⋅292^{4} \cdot 29
Sign: −0.485+0.874i-0.485 + 0.874i
Analytic conductor: 3.705053.70505
Root analytic conductor: 1.924851.92485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ464(367,⋅)\chi_{464} (367, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 464, ( :1/2), −0.485+0.874i)(2,\ 464,\ (\ :1/2),\ -0.485 + 0.874i)

Particular Values

L(1)L(1) ≈\approx 0.404465−0.687161i0.404465 - 0.687161i
L(12)L(\frac12) ≈\approx 0.404465−0.687161i0.404465 - 0.687161i
L(32)L(\frac{3}{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
29 1+(−3.03−4.44i)T 1 + (-3.03 - 4.44i)T
good3 1+(−0.432−0.151i)T+(2.34+1.87i)T2 1 + (-0.432 - 0.151i)T + (2.34 + 1.87i)T^{2}
5 1+(1.40+0.320i)T+(4.50+2.16i)T2 1 + (1.40 + 0.320i)T + (4.50 + 2.16i)T^{2}
7 1+(0.759+1.57i)T+(−4.36+5.47i)T2 1 + (0.759 + 1.57i)T + (-4.36 + 5.47i)T^{2}
11 1+(−0.426+3.78i)T+(−10.7−2.44i)T2 1 + (-0.426 + 3.78i)T + (-10.7 - 2.44i)T^{2}
13 1+(3.71−2.96i)T+(2.89−12.6i)T2 1 + (3.71 - 2.96i)T + (2.89 - 12.6i)T^{2}
17 1+(−1.08+1.08i)T−17iT2 1 + (-1.08 + 1.08i)T - 17iT^{2}
19 1+(1.57+4.48i)T+(−14.8+11.8i)T2 1 + (1.57 + 4.48i)T + (-14.8 + 11.8i)T^{2}
23 1+(0.0809−0.0184i)T+(20.7−9.97i)T2 1 + (0.0809 - 0.0184i)T + (20.7 - 9.97i)T^{2}
31 1+(−8.91+5.60i)T+(13.4−27.9i)T2 1 + (-8.91 + 5.60i)T + (13.4 - 27.9i)T^{2}
37 1+(5.41−0.610i)T+(36.0−8.23i)T2 1 + (5.41 - 0.610i)T + (36.0 - 8.23i)T^{2}
41 1+(0.943+0.943i)T+41iT2 1 + (0.943 + 0.943i)T + 41iT^{2}
43 1+(−0.824+1.31i)T+(−18.6−38.7i)T2 1 + (-0.824 + 1.31i)T + (-18.6 - 38.7i)T^{2}
47 1+(6.23+0.702i)T+(45.8+10.4i)T2 1 + (6.23 + 0.702i)T + (45.8 + 10.4i)T^{2}
53 1+(1.20−5.27i)T+(−47.7−22.9i)T2 1 + (1.20 - 5.27i)T + (-47.7 - 22.9i)T^{2}
59 1−13.1iT−59T2 1 - 13.1iT - 59T^{2}
61 1+(0.0136−0.0389i)T+(−47.6−38.0i)T2 1 + (0.0136 - 0.0389i)T + (-47.6 - 38.0i)T^{2}
67 1+(−5.77+7.24i)T+(−14.9−65.3i)T2 1 + (-5.77 + 7.24i)T + (-14.9 - 65.3i)T^{2}
71 1+(2.67+3.35i)T+(−15.7+69.2i)T2 1 + (2.67 + 3.35i)T + (-15.7 + 69.2i)T^{2}
73 1+(−7.63+12.1i)T+(−31.6−65.7i)T2 1 + (-7.63 + 12.1i)T + (-31.6 - 65.7i)T^{2}
79 1+(−5.46+0.616i)T+(77.0−17.5i)T2 1 + (-5.46 + 0.616i)T + (77.0 - 17.5i)T^{2}
83 1+(5.03−10.4i)T+(−51.7−64.8i)T2 1 + (5.03 - 10.4i)T + (-51.7 - 64.8i)T^{2}
89 1+(2.84+4.52i)T+(−38.6+80.1i)T2 1 + (2.84 + 4.52i)T + (-38.6 + 80.1i)T^{2}
97 1+(−3.29−9.40i)T+(−75.8+60.4i)T2 1 + (-3.29 - 9.40i)T + (-75.8 + 60.4i)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

−10.83425052977644722193873730974, −9.752449636299469113047579045908, −8.905853578243021280555464406093, −8.135154281934619845829127970425, −7.05042101926919263954610604592, −6.19449877650818291392112407579, −4.81573573652917167622783583542, −3.77248655630894694113549958077, −2.73864005154981613113038842211, −0.45548456559717816370641086771, 2.17953417817838588019758023822, 3.26194855317310529326761037321, 4.64210740826495191354345563455, 5.63085636206256354259704971809, 6.83294890190938577140423049001, 7.922522379946927416530586315594, 8.341252884944656620959997678714, 9.726157381873214768166074904786, 10.26050156583128793508453555152, 11.49791357624566246560267796985

Graph of the ZZ-function along the critical line