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Properties

Label 2-665-19.11-c1-0-35
Degree 22
Conductor 665665
Sign −0.915+0.403i-0.915 + 0.403i
Analytic cond. 5.310055.31005
Root an. cond. 2.304352.30435
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  + (0.745 − 1.29i)2-s + (1.62 − 2.82i)3-s + (−0.112 − 0.194i)4-s + (0.5 − 0.866i)5-s + (−2.42 − 4.20i)6-s − 7-s + 2.64·8-s + (−3.80 − 6.59i)9-s + (−0.745 − 1.29i)10-s − 4.16·11-s − 0.730·12-s + (2.44 + 4.24i)13-s + (−0.745 + 1.29i)14-s + (−1.62 − 2.82i)15-s + (2.19 − 3.80i)16-s + (−1.32 + 2.29i)17-s + ⋯
L(s)  = 1  + (0.527 − 0.913i)2-s + (0.940 − 1.62i)3-s + (−0.0560 − 0.0970i)4-s + (0.223 − 0.387i)5-s + (−0.991 − 1.71i)6-s − 0.377·7-s + 0.936·8-s + (−1.26 − 2.19i)9-s + (−0.235 − 0.408i)10-s − 1.25·11-s − 0.210·12-s + (0.679 + 1.17i)13-s + (−0.199 + 0.345i)14-s + (−0.420 − 0.728i)15-s + (0.549 − 0.952i)16-s + (−0.321 + 0.556i)17-s + ⋯

Functional equation

Λ(s)=(665s/2 ΓC(s) L(s)=((−0.915+0.403i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(665s/2 ΓC(s+1/2) L(s)=((−0.915+0.403i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 665665    =    5⋅7⋅195 \cdot 7 \cdot 19
Sign: −0.915+0.403i-0.915 + 0.403i
Analytic conductor: 5.310055.31005
Root analytic conductor: 2.304352.30435
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ665(106,⋅)\chi_{665} (106, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 665, ( :1/2), −0.915+0.403i)(2,\ 665,\ (\ :1/2),\ -0.915 + 0.403i)

Particular Values

L(1)L(1) ≈\approx 0.566245−2.68881i0.566245 - 2.68881i
L(12)L(\frac12) ≈\approx 0.566245−2.68881i0.566245 - 2.68881i
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)
bad5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+T 1 + T
19 1+(−4.27+0.875i)T 1 + (-4.27 + 0.875i)T
good2 1+(−0.745+1.29i)T+(−1−1.73i)T2 1 + (-0.745 + 1.29i)T + (-1 - 1.73i)T^{2}
3 1+(−1.62+2.82i)T+(−1.5−2.59i)T2 1 + (-1.62 + 2.82i)T + (-1.5 - 2.59i)T^{2}
11 1+4.16T+11T2 1 + 4.16T + 11T^{2}
13 1+(−2.44−4.24i)T+(−6.5+11.2i)T2 1 + (-2.44 - 4.24i)T + (-6.5 + 11.2i)T^{2}
17 1+(1.32−2.29i)T+(−8.5−14.7i)T2 1 + (1.32 - 2.29i)T + (-8.5 - 14.7i)T^{2}
23 1+(1.11+1.92i)T+(−11.5+19.9i)T2 1 + (1.11 + 1.92i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.65−4.59i)T+(−14.5+25.1i)T2 1 + (-2.65 - 4.59i)T + (-14.5 + 25.1i)T^{2}
31 1+0.565T+31T2 1 + 0.565T + 31T^{2}
37 1−5.73T+37T2 1 - 5.73T + 37T^{2}
41 1+(−1.06+1.84i)T+(−20.5−35.5i)T2 1 + (-1.06 + 1.84i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.07+5.32i)T+(−21.5−37.2i)T2 1 + (-3.07 + 5.32i)T + (-21.5 - 37.2i)T^{2}
47 1+(2.75+4.77i)T+(−23.5+40.7i)T2 1 + (2.75 + 4.77i)T + (-23.5 + 40.7i)T^{2}
53 1+(−5.04−8.74i)T+(−26.5+45.8i)T2 1 + (-5.04 - 8.74i)T + (-26.5 + 45.8i)T^{2}
59 1+(3.35−5.80i)T+(−29.5−51.0i)T2 1 + (3.35 - 5.80i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.14+7.18i)T+(−30.5+52.8i)T2 1 + (4.14 + 7.18i)T + (-30.5 + 52.8i)T^{2}
67 1+(−4.92−8.53i)T+(−33.5+58.0i)T2 1 + (-4.92 - 8.53i)T + (-33.5 + 58.0i)T^{2}
71 1+(7.43−12.8i)T+(−35.5−61.4i)T2 1 + (7.43 - 12.8i)T + (-35.5 - 61.4i)T^{2}
73 1+(3.17−5.49i)T+(−36.5−63.2i)T2 1 + (3.17 - 5.49i)T + (-36.5 - 63.2i)T^{2}
79 1+(−3.09+5.36i)T+(−39.5−68.4i)T2 1 + (-3.09 + 5.36i)T + (-39.5 - 68.4i)T^{2}
83 1+13.7T+83T2 1 + 13.7T + 83T^{2}
89 1+(−4.99−8.65i)T+(−44.5+77.0i)T2 1 + (-4.99 - 8.65i)T + (-44.5 + 77.0i)T^{2}
97 1+(−9.47+16.4i)T+(−48.5−84.0i)T2 1 + (-9.47 + 16.4i)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

−10.27836588336695200106611273087, −9.093318267255837581347657299942, −8.386811289806540105272123518222, −7.49931598268468800907520251144, −6.80095943414990803579585450576, −5.66753545292714829555951230135, −4.15567563110485461862514983084, −3.01967251429711354642267533906, −2.26783404473652878376029297595, −1.24754237334955688559499521969, 2.62986342130104868089953086044, 3.43675200694931319290719701908, 4.61757711540811727933213765344, 5.37074268846602692314519149579, 6.11501099306184418282726091543, 7.70686175548981804967795273275, 8.001173460853755351638394221863, 9.302041911217551512565245623105, 10.05892937034877592975743152435, 10.54893829780075620300505068725

Graph of the ZZ-function along the critical line