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Properties

Label 2-416-104.101-c1-0-5
Degree 22
Conductor 416416
Sign 0.862−0.505i0.862 - 0.505i
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  + (−0.633 + 0.366i)3-s + 3.73·5-s + (3 + 1.73i)7-s + (−1.23 + 2.13i)9-s + (−1 − 1.73i)11-s + (−2.59 − 2.5i)13-s + (−2.36 + 1.36i)15-s + (0.232 − 0.401i)17-s + (−0.633 + 1.09i)19-s − 2.53·21-s + (4.09 + 7.09i)23-s + 8.92·25-s − 4i·27-s + (2.59 − 1.5i)29-s − 4.73i·31-s + ⋯
L(s)  = 1  + (−0.366 + 0.211i)3-s + 1.66·5-s + (1.13 + 0.654i)7-s + (−0.410 + 0.711i)9-s + (−0.301 − 0.522i)11-s + (−0.720 − 0.693i)13-s + (−0.610 + 0.352i)15-s + (0.0562 − 0.0974i)17-s + (−0.145 + 0.251i)19-s − 0.553·21-s + (0.854 + 1.48i)23-s + 1.78·25-s − 0.769i·27-s + (0.482 − 0.278i)29-s − 0.849i·31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.862−0.505i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.862−0.505i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.862−0.505i0.862 - 0.505i
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), 0.862−0.505i)(2,\ 416,\ (\ :1/2),\ 0.862 - 0.505i)

Particular Values

L(1)L(1) ≈\approx 1.61454+0.438169i1.61454 + 0.438169i
L(12)L(\frac12) ≈\approx 1.61454+0.438169i1.61454 + 0.438169i
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+(0.633−0.366i)T+(1.5−2.59i)T2 1 + (0.633 - 0.366i)T + (1.5 - 2.59i)T^{2}
5 1−3.73T+5T2 1 - 3.73T + 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+(−0.232+0.401i)T+(−8.5−14.7i)T2 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.633−1.09i)T+(−9.5−16.4i)T2 1 + (0.633 - 1.09i)T + (-9.5 - 16.4i)T^{2}
23 1+(−4.09−7.09i)T+(−11.5+19.9i)T2 1 + (-4.09 - 7.09i)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+4.73iT−31T2 1 + 4.73iT - 31T^{2}
37 1+(−2.13−3.69i)T+(−18.5+32.0i)T2 1 + (-2.13 - 3.69i)T + (-18.5 + 32.0i)T^{2}
41 1+(7.96−4.59i)T+(20.5−35.5i)T2 1 + (7.96 - 4.59i)T + (20.5 - 35.5i)T^{2}
43 1+(2.19+1.26i)T+(21.5+37.2i)T2 1 + (2.19 + 1.26i)T + (21.5 + 37.2i)T^{2}
47 1+6.73iT−47T2 1 + 6.73iT - 47T^{2}
53 1+3.92iT−53T2 1 + 3.92iT - 53T^{2}
59 1+(0.267−0.464i)T+(−29.5−51.0i)T2 1 + (0.267 - 0.464i)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+(3.63+6.29i)T+(−33.5+58.0i)T2 1 + (3.63 + 6.29i)T + (-33.5 + 58.0i)T^{2}
71 1+(8.02+4.63i)T+(35.5+61.4i)T2 1 + (8.02 + 4.63i)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+1.46T+83T2 1 + 1.46T + 83T^{2}
89 1+(−6.46+3.73i)T+(44.5−77.0i)T2 1 + (-6.46 + 3.73i)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

−11.21179342211686432477877550922, −10.35615756214236753193753155132, −9.641412388378613546966654067472, −8.576882916538603149633808579381, −7.74558994094260016705673428566, −6.23484463804451982473609812478, −5.25776138198516549859456139342, −5.12218582563962173550692513912, −2.80573823234163488842739810010, −1.77014196261756404392811871620, 1.37689456409559378996232965991, 2.58228666330642898521846898328, 4.57670811670876665145039686853, 5.31537771085796384898704365689, 6.47880135829166454267554200066, 7.12750031486855883641796761728, 8.563026409030889233142383571549, 9.368806809361251483086610948737, 10.33489965639451944844706599530, 10.95504223102545002022818735892

Graph of the ZZ-function along the critical line