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Properties

Label 2-416-13.3-c1-0-13
Degree 22
Conductor 416416
Sign −0.923−0.384i-0.923 - 0.384i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
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  + (−1.17 − 2.04i)3-s − 2.56·5-s + (1.84 − 3.18i)7-s + (−1.28 + 2.21i)9-s + (0.516 + 0.895i)11-s + (−2.84 − 2.21i)13-s + (3.02 + 5.23i)15-s + (−3.06 + 5.30i)17-s + (−2.50 + 4.33i)19-s − 8.68·21-s + (−3.53 − 6.12i)23-s + 1.56·25-s − 1.03·27-s + (2.5 + 4.33i)29-s + 3.39·31-s + ⋯
L(s)  = 1  + (−0.680 − 1.17i)3-s − 1.14·5-s + (0.695 − 1.20i)7-s + (−0.426 + 0.739i)9-s + (0.155 + 0.269i)11-s + (−0.788 − 0.615i)13-s + (0.779 + 1.35i)15-s + (−0.742 + 1.28i)17-s + (−0.574 + 0.994i)19-s − 1.89·21-s + (−0.737 − 1.27i)23-s + 0.312·25-s − 0.198·27-s + (0.464 + 0.804i)29-s + 0.609·31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.923−0.384i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.923−0.384i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.923−0.384i-0.923 - 0.384i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(289,⋅)\chi_{416} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.923−0.384i)(2,\ 416,\ (\ :1/2),\ -0.923 - 0.384i)

Particular Values

L(1)L(1) ≈\approx 0.0786992+0.393783i0.0786992 + 0.393783i
L(12)L(\frac12) ≈\approx 0.0786992+0.393783i0.0786992 + 0.393783i
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
13 1+(2.84+2.21i)T 1 + (2.84 + 2.21i)T
good3 1+(1.17+2.04i)T+(−1.5+2.59i)T2 1 + (1.17 + 2.04i)T + (-1.5 + 2.59i)T^{2}
5 1+2.56T+5T2 1 + 2.56T + 5T^{2}
7 1+(−1.84+3.18i)T+(−3.5−6.06i)T2 1 + (-1.84 + 3.18i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.516−0.895i)T+(−5.5+9.52i)T2 1 + (-0.516 - 0.895i)T + (-5.5 + 9.52i)T^{2}
17 1+(3.06−5.30i)T+(−8.5−14.7i)T2 1 + (3.06 - 5.30i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.50−4.33i)T+(−9.5−16.4i)T2 1 + (2.50 - 4.33i)T + (-9.5 - 16.4i)T^{2}
23 1+(3.53+6.12i)T+(−11.5+19.9i)T2 1 + (3.53 + 6.12i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.5−4.33i)T+(−14.5+25.1i)T2 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2}
31 1−3.39T+31T2 1 - 3.39T + 31T^{2}
37 1+(1.06+1.83i)T+(−18.5+32.0i)T2 1 + (1.06 + 1.83i)T + (-18.5 + 32.0i)T^{2}
41 1+(−2.06−3.57i)T+(−20.5+35.5i)T2 1 + (-2.06 - 3.57i)T + (-20.5 + 35.5i)T^{2}
43 1+(−4.19+7.27i)T+(−21.5−37.2i)T2 1 + (-4.19 + 7.27i)T + (-21.5 - 37.2i)T^{2}
47 1+10.7T+47T2 1 + 10.7T + 47T^{2}
53 1+2.56T+53T2 1 + 2.56T + 53T^{2}
59 1+(−5.23+9.06i)T+(−29.5−51.0i)T2 1 + (-5.23 + 9.06i)T + (-29.5 - 51.0i)T^{2}
61 1+(−5.62+9.73i)T+(−30.5−52.8i)T2 1 + (-5.62 + 9.73i)T + (-30.5 - 52.8i)T^{2}
67 1+(4.86+8.42i)T+(−33.5+58.0i)T2 1 + (4.86 + 8.42i)T + (-33.5 + 58.0i)T^{2}
71 1+(−2.50+4.33i)T+(−35.5−61.4i)T2 1 + (-2.50 + 4.33i)T + (-35.5 - 61.4i)T^{2}
73 1+4.31T+73T2 1 + 4.31T + 73T^{2}
79 1+11.5T+79T2 1 + 11.5T + 79T^{2}
83 1−2.64T+83T2 1 - 2.64T + 83T^{2}
89 1+(5.34+9.25i)T+(−44.5+77.0i)T2 1 + (5.34 + 9.25i)T + (-44.5 + 77.0i)T^{2}
97 1+(−4.90+8.49i)T+(−48.5−84.0i)T2 1 + (-4.90 + 8.49i)T + (-48.5 - 84.0i)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

−10.84543053470160466159728703681, −10.20279867674266124495607028120, −8.271909493027483777213275579954, −7.902661828663575642050737003162, −7.01092573626809422560437551415, −6.23820512081728087858916907981, −4.70628281085547355696063768583, −3.84897654514970693305286304510, −1.78643903968965349693797247115, −0.27654549078756125280518451938, 2.60188210447576842164962080250, 4.21216447585704202026555726346, 4.76044385149721763951763037962, 5.72765656043330839141487503292, 7.08879894756141722930873384655, 8.203263839334334264930794387713, 9.124955677534292119769525893900, 9.862487893835154234019996276761, 11.17256101563388452500090453539, 11.65876507924949341170271558566

Graph of the ZZ-function along the critical line