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Properties

Label 2-650-25.11-c1-0-2
Degree 22
Conductor 650650
Sign −0.0206−0.999i-0.0206 - 0.999i
Analytic cond. 5.190275.19027
Root an. cond. 2.278212.27821
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  + (0.809 − 0.587i)2-s + (−0.311 + 0.960i)3-s + (0.309 − 0.951i)4-s + (−2.22 + 0.233i)5-s + (0.311 + 0.960i)6-s − 0.400·7-s + (−0.309 − 0.951i)8-s + (1.60 + 1.16i)9-s + (−1.66 + 1.49i)10-s + (−3.50 + 2.54i)11-s + (0.816 + 0.593i)12-s + (0.809 + 0.587i)13-s + (−0.324 + 0.235i)14-s + (0.469 − 2.20i)15-s + (−0.809 − 0.587i)16-s + (1.02 + 3.14i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.180 + 0.554i)3-s + (0.154 − 0.475i)4-s + (−0.994 + 0.104i)5-s + (0.127 + 0.391i)6-s − 0.151·7-s + (−0.109 − 0.336i)8-s + (0.534 + 0.388i)9-s + (−0.525 + 0.473i)10-s + (−1.05 + 0.766i)11-s + (0.235 + 0.171i)12-s + (0.224 + 0.163i)13-s + (−0.0866 + 0.0629i)14-s + (0.121 − 0.570i)15-s + (−0.202 − 0.146i)16-s + (0.248 + 0.763i)17-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((−0.0206−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+1/2) L(s)=((−0.0206−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: −0.0206−0.999i-0.0206 - 0.999i
Analytic conductor: 5.190275.19027
Root analytic conductor: 2.278212.27821
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ650(261,⋅)\chi_{650} (261, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :1/2), −0.0206−0.999i)(2,\ 650,\ (\ :1/2),\ -0.0206 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.816107+0.833165i0.816107 + 0.833165i
L(12)L(\frac12) ≈\approx 0.816107+0.833165i0.816107 + 0.833165i
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+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
5 1+(2.22−0.233i)T 1 + (2.22 - 0.233i)T
13 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
good3 1+(0.311−0.960i)T+(−2.42−1.76i)T2 1 + (0.311 - 0.960i)T + (-2.42 - 1.76i)T^{2}
7 1+0.400T+7T2 1 + 0.400T + 7T^{2}
11 1+(3.50−2.54i)T+(3.39−10.4i)T2 1 + (3.50 - 2.54i)T + (3.39 - 10.4i)T^{2}
17 1+(−1.02−3.14i)T+(−13.7+9.99i)T2 1 + (-1.02 - 3.14i)T + (-13.7 + 9.99i)T^{2}
19 1+(−1.63−5.01i)T+(−15.3+11.1i)T2 1 + (-1.63 - 5.01i)T + (-15.3 + 11.1i)T^{2}
23 1+(4.42−3.21i)T+(7.10−21.8i)T2 1 + (4.42 - 3.21i)T + (7.10 - 21.8i)T^{2}
29 1+(2.40−7.38i)T+(−23.4−17.0i)T2 1 + (2.40 - 7.38i)T + (-23.4 - 17.0i)T^{2}
31 1+(2.54+7.82i)T+(−25.0+18.2i)T2 1 + (2.54 + 7.82i)T + (-25.0 + 18.2i)T^{2}
37 1+(3.58+2.60i)T+(11.4+35.1i)T2 1 + (3.58 + 2.60i)T + (11.4 + 35.1i)T^{2}
41 1+(−7.60−5.52i)T+(12.6+38.9i)T2 1 + (-7.60 - 5.52i)T + (12.6 + 38.9i)T^{2}
43 1−6.47T+43T2 1 - 6.47T + 43T^{2}
47 1+(−0.115+0.356i)T+(−38.0−27.6i)T2 1 + (-0.115 + 0.356i)T + (-38.0 - 27.6i)T^{2}
53 1+(−0.968+2.98i)T+(−42.8−31.1i)T2 1 + (-0.968 + 2.98i)T + (-42.8 - 31.1i)T^{2}
59 1+(−2.12−1.54i)T+(18.2+56.1i)T2 1 + (-2.12 - 1.54i)T + (18.2 + 56.1i)T^{2}
61 1+(6.95−5.05i)T+(18.8−58.0i)T2 1 + (6.95 - 5.05i)T + (18.8 - 58.0i)T^{2}
67 1+(3.29+10.1i)T+(−54.2+39.3i)T2 1 + (3.29 + 10.1i)T + (-54.2 + 39.3i)T^{2}
71 1+(−0.974+3.00i)T+(−57.4−41.7i)T2 1 + (-0.974 + 3.00i)T + (-57.4 - 41.7i)T^{2}
73 1+(−11.6+8.48i)T+(22.5−69.4i)T2 1 + (-11.6 + 8.48i)T + (22.5 - 69.4i)T^{2}
79 1+(2.55−7.86i)T+(−63.9−46.4i)T2 1 + (2.55 - 7.86i)T + (-63.9 - 46.4i)T^{2}
83 1+(0.171+0.527i)T+(−67.1+48.7i)T2 1 + (0.171 + 0.527i)T + (-67.1 + 48.7i)T^{2}
89 1+(−14.5+10.5i)T+(27.5−84.6i)T2 1 + (-14.5 + 10.5i)T + (27.5 - 84.6i)T^{2}
97 1+(0.961−2.95i)T+(−78.4−57.0i)T2 1 + (0.961 - 2.95i)T + (-78.4 - 57.0i)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

−10.75013992573974074318447328762, −10.20307825566116416208055556871, −9.331247734140510691071501473271, −7.84212105097278781213080738023, −7.48805544241332256636673071782, −6.02328791666546400016374970726, −5.06394625402122012618952716393, −4.14182482325432704870577391854, −3.45477044361098070239073559906, −1.87684819629777063975174580453, 0.52945545925845163484689775164, 2.70714883165405939312202811560, 3.77804168583553688533509583824, 4.82712538556869316303694159768, 5.83762751958281168714862912496, 6.86683587144083810990552586576, 7.55737698938383873109375563342, 8.249706188703391956507506914422, 9.300412942757988391715295910710, 10.58601441165867234054146032477

Graph of the ZZ-function along the critical line