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Properties

Label 2-416-104.101-c1-0-9
Degree 22
Conductor 416416
Sign 0.869+0.494i0.869 + 0.494i
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  + (2.36 − 1.36i)3-s − 0.267·5-s + (3 + 1.73i)7-s + (2.23 − 3.86i)9-s + (1 + 1.73i)11-s + (−2.59 − 2.5i)13-s + (−0.633 + 0.366i)15-s + (−3.23 + 5.59i)17-s + (2.36 − 4.09i)19-s + 9.46·21-s + (−1.09 − 1.90i)23-s − 4.92·25-s − 4.00i·27-s + (2.59 − 1.5i)29-s + 1.26i·31-s + ⋯
L(s)  = 1  + (1.36 − 0.788i)3-s − 0.119·5-s + (1.13 + 0.654i)7-s + (0.744 − 1.28i)9-s + (0.301 + 0.522i)11-s + (−0.720 − 0.693i)13-s + (−0.163 + 0.0945i)15-s + (−0.783 + 1.35i)17-s + (0.542 − 0.940i)19-s + 2.06·21-s + (−0.228 − 0.396i)23-s − 0.985·25-s − 0.769i·27-s + (0.482 − 0.278i)29-s + 0.227i·31-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.869+0.494i0.869 + 0.494i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(49,⋅)\chi_{416} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.869+0.494i)(2,\ 416,\ (\ :1/2),\ 0.869 + 0.494i)

Particular Values

L(1)L(1) ≈\approx 2.14387−0.567088i2.14387 - 0.567088i
L(12)L(\frac12) ≈\approx 2.14387−0.567088i2.14387 - 0.567088i
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+(2.59+2.5i)T 1 + (2.59 + 2.5i)T
good3 1+(−2.36+1.36i)T+(1.5−2.59i)T2 1 + (-2.36 + 1.36i)T + (1.5 - 2.59i)T^{2}
5 1+0.267T+5T2 1 + 0.267T + 5T^{2}
7 1+(−3−1.73i)T+(3.5+6.06i)T2 1 + (-3 - 1.73i)T + (3.5 + 6.06i)T^{2}
11 1+(−1−1.73i)T+(−5.5+9.52i)T2 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2}
17 1+(3.23−5.59i)T+(−8.5−14.7i)T2 1 + (3.23 - 5.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.36+4.09i)T+(−9.5−16.4i)T2 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.09+1.90i)T+(−11.5+19.9i)T2 1 + (1.09 + 1.90i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.59+1.5i)T+(14.5−25.1i)T2 1 + (-2.59 + 1.5i)T + (14.5 - 25.1i)T^{2}
31 1−1.26iT−31T2 1 - 1.26iT - 31T^{2}
37 1+(3.86+6.69i)T+(−18.5+32.0i)T2 1 + (3.86 + 6.69i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.03−0.598i)T+(20.5−35.5i)T2 1 + (1.03 - 0.598i)T + (20.5 - 35.5i)T^{2}
43 1+(8.19+4.73i)T+(21.5+37.2i)T2 1 + (8.19 + 4.73i)T + (21.5 + 37.2i)T^{2}
47 1−3.26iT−47T2 1 - 3.26iT - 47T^{2}
53 1−9.92iT−53T2 1 - 9.92iT - 53T^{2}
59 1+(−3.73+6.46i)T+(−29.5−51.0i)T2 1 + (-3.73 + 6.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.866−0.5i)T+(30.5+52.8i)T2 1 + (-0.866 - 0.5i)T + (30.5 + 52.8i)T^{2}
67 1+(−5.36−9.29i)T+(−33.5+58.0i)T2 1 + (-5.36 - 9.29i)T + (-33.5 + 58.0i)T^{2}
71 1+(−11.0−6.36i)T+(35.5+61.4i)T2 1 + (-11.0 - 6.36i)T + (35.5 + 61.4i)T^{2}
73 1+1.73iT−73T2 1 + 1.73iT - 73T^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1+5.46T+83T2 1 + 5.46T + 83T^{2}
89 1+(0.464−0.267i)T+(44.5−77.0i)T2 1 + (0.464 - 0.267i)T + (44.5 - 77.0i)T^{2}
97 1+(−5.19−3i)T+(48.5+84.0i)T2 1 + (-5.19 - 3i)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

−11.27762376141748608947925103301, −10.06893405464887217921104285338, −8.970989151746805224911745052336, −8.346373009948488024902477988936, −7.67534564828097902615337365244, −6.74743233091417246276279002591, −5.30055637005559759203860821303, −4.04392846819313277584010069805, −2.58420996050725076309309086862, −1.76629636397938933427033842376, 1.89807204594735360958070209626, 3.28787758266785757970085005025, 4.27949759026825404671493706634, 5.08458303967688827187535036294, 6.91078046821121709237118430380, 7.88657552138808743317443178834, 8.494390710908247392814130186058, 9.528093481241718164448345801617, 10.07419271348816401205714715288, 11.32003498994018216089960939791

Graph of the ZZ-function along the critical line