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Properties

Label 2-464-29.22-c1-0-3
Degree 22
Conductor 464464
Sign 0.596−0.802i0.596 - 0.802i
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  + (0.186 − 0.386i)3-s + (0.478 + 2.09i)5-s + (1.33 + 0.641i)7-s + (1.75 + 2.20i)9-s + (−2.02 − 1.61i)11-s + (−1.55 + 1.94i)13-s + (0.899 + 0.205i)15-s + 2.05i·17-s + (−0.289 − 0.600i)19-s + (0.496 − 0.395i)21-s + (−1.34 + 5.89i)23-s + (0.340 − 0.163i)25-s + (2.43 − 0.555i)27-s + (5.10 + 1.70i)29-s + (2.83 − 0.647i)31-s + ⋯
L(s)  = 1  + (0.107 − 0.223i)3-s + (0.213 + 0.937i)5-s + (0.503 + 0.242i)7-s + (0.585 + 0.733i)9-s + (−0.611 − 0.487i)11-s + (−0.430 + 0.539i)13-s + (0.232 + 0.0530i)15-s + 0.499i·17-s + (−0.0663 − 0.137i)19-s + (0.108 − 0.0863i)21-s + (−0.280 + 1.22i)23-s + (0.0681 − 0.0327i)25-s + (0.468 − 0.106i)27-s + (0.948 + 0.317i)29-s + (0.509 − 0.116i)31-s + ⋯

Functional equation

Λ(s)=(464s/2 ΓC(s) L(s)=((0.596−0.802i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(464s/2 ΓC(s+1/2) L(s)=((0.596−0.802i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 464464    =    24⋅292^{4} \cdot 29
Sign: 0.596−0.802i0.596 - 0.802i
Analytic conductor: 3.705053.70505
Root analytic conductor: 1.924851.92485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ464(225,⋅)\chi_{464} (225, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 464, ( :1/2), 0.596−0.802i)(2,\ 464,\ (\ :1/2),\ 0.596 - 0.802i)

Particular Values

L(1)L(1) ≈\approx 1.37425+0.690846i1.37425 + 0.690846i
L(12)L(\frac12) ≈\approx 1.37425+0.690846i1.37425 + 0.690846i
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+(−5.10−1.70i)T 1 + (-5.10 - 1.70i)T
good3 1+(−0.186+0.386i)T+(−1.87−2.34i)T2 1 + (-0.186 + 0.386i)T + (-1.87 - 2.34i)T^{2}
5 1+(−0.478−2.09i)T+(−4.50+2.16i)T2 1 + (-0.478 - 2.09i)T + (-4.50 + 2.16i)T^{2}
7 1+(−1.33−0.641i)T+(4.36+5.47i)T2 1 + (-1.33 - 0.641i)T + (4.36 + 5.47i)T^{2}
11 1+(2.02+1.61i)T+(2.44+10.7i)T2 1 + (2.02 + 1.61i)T + (2.44 + 10.7i)T^{2}
13 1+(1.55−1.94i)T+(−2.89−12.6i)T2 1 + (1.55 - 1.94i)T + (-2.89 - 12.6i)T^{2}
17 1−2.05iT−17T2 1 - 2.05iT - 17T^{2}
19 1+(0.289+0.600i)T+(−11.8+14.8i)T2 1 + (0.289 + 0.600i)T + (-11.8 + 14.8i)T^{2}
23 1+(1.34−5.89i)T+(−20.7−9.97i)T2 1 + (1.34 - 5.89i)T + (-20.7 - 9.97i)T^{2}
31 1+(−2.83+0.647i)T+(27.9−13.4i)T2 1 + (-2.83 + 0.647i)T + (27.9 - 13.4i)T^{2}
37 1+(−2.53+2.01i)T+(8.23−36.0i)T2 1 + (-2.53 + 2.01i)T + (8.23 - 36.0i)T^{2}
41 1+6.01iT−41T2 1 + 6.01iT - 41T^{2}
43 1+(−2.70−0.617i)T+(38.7+18.6i)T2 1 + (-2.70 - 0.617i)T + (38.7 + 18.6i)T^{2}
47 1+(−8.30−6.62i)T+(10.4+45.8i)T2 1 + (-8.30 - 6.62i)T + (10.4 + 45.8i)T^{2}
53 1+(2.47+10.8i)T+(−47.7+22.9i)T2 1 + (2.47 + 10.8i)T + (-47.7 + 22.9i)T^{2}
59 1+12.2T+59T2 1 + 12.2T + 59T^{2}
61 1+(−0.151+0.314i)T+(−38.0−47.6i)T2 1 + (-0.151 + 0.314i)T + (-38.0 - 47.6i)T^{2}
67 1+(6.06+7.60i)T+(−14.9+65.3i)T2 1 + (6.06 + 7.60i)T + (-14.9 + 65.3i)T^{2}
71 1+(1.57−1.97i)T+(−15.7−69.2i)T2 1 + (1.57 - 1.97i)T + (-15.7 - 69.2i)T^{2}
73 1+(−4.10−0.937i)T+(65.7+31.6i)T2 1 + (-4.10 - 0.937i)T + (65.7 + 31.6i)T^{2}
79 1+(−4.21+3.35i)T+(17.5−77.0i)T2 1 + (-4.21 + 3.35i)T + (17.5 - 77.0i)T^{2}
83 1+(7.37−3.55i)T+(51.7−64.8i)T2 1 + (7.37 - 3.55i)T + (51.7 - 64.8i)T^{2}
89 1+(−0.477+0.109i)T+(80.1−38.6i)T2 1 + (-0.477 + 0.109i)T + (80.1 - 38.6i)T^{2}
97 1+(1.92+3.99i)T+(−60.4+75.8i)T2 1 + (1.92 + 3.99i)T + (-60.4 + 75.8i)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

−10.97286246706472074430269177954, −10.48395929314594763786800725215, −9.484519074831811551734109443456, −8.276316126920630314518216719271, −7.53906936474832081720305317109, −6.64946522314782897327199422720, −5.54559931189942773325961372334, −4.44305822355128969355693103591, −2.95027085289615429178139260314, −1.87346378385879319836862778137, 1.03097704656593494295862094795, 2.72330140501008922941198282442, 4.39707623157990257867474568403, 4.88073362466384477139287000866, 6.18053766490030872431942925281, 7.36831255734812304391696604937, 8.251460567921053834327821427158, 9.159982848146591571784807020421, 10.00449416131714486511383482351, 10.69419471517765078591680301852

Graph of the ZZ-function along the critical line