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Properties

Label 2-2664-2664.1627-c0-0-8
Degree 22
Conductor 26642664
Sign −0.997−0.0697i-0.997 - 0.0697i
Analytic cond. 1.329501.32950
Root an. cond. 1.153041.15304
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.309 + 0.951i)3-s + (−0.499 − 0.866i)4-s + (−0.809 − 1.40i)5-s + (0.978 + 0.207i)6-s − 0.999·8-s + (−0.809 + 0.587i)9-s − 1.61·10-s + (0.104 − 0.181i)11-s + (0.669 − 0.743i)12-s + (−0.978 − 1.69i)13-s + (1.08 − 1.20i)15-s + (−0.5 + 0.866i)16-s + (0.104 + 0.994i)18-s + (−0.809 + 1.40i)20-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.309 + 0.951i)3-s + (−0.499 − 0.866i)4-s + (−0.809 − 1.40i)5-s + (0.978 + 0.207i)6-s − 0.999·8-s + (−0.809 + 0.587i)9-s − 1.61·10-s + (0.104 − 0.181i)11-s + (0.669 − 0.743i)12-s + (−0.978 − 1.69i)13-s + (1.08 − 1.20i)15-s + (−0.5 + 0.866i)16-s + (0.104 + 0.994i)18-s + (−0.809 + 1.40i)20-s + ⋯

Functional equation

Λ(s)=(2664s/2 ΓC(s) L(s)=((−0.997−0.0697i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0697i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2664s/2 ΓC(s) L(s)=((−0.997−0.0697i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0697i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26642664    =    23⋅32⋅372^{3} \cdot 3^{2} \cdot 37
Sign: −0.997−0.0697i-0.997 - 0.0697i
Analytic conductor: 1.329501.32950
Root analytic conductor: 1.153041.15304
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2664(1627,⋅)\chi_{2664} (1627, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2664, ( :0), −0.997−0.0697i)(2,\ 2664,\ (\ :0),\ -0.997 - 0.0697i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.73774778790.7377477879
L(12)L(\frac12) ≈\approx 0.73774778790.7377477879
L(1)L(1) 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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
37 1+T 1 + T
good5 1+(0.809+1.40i)T+(−0.5+0.866i)T2 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2}
7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(−0.104+0.181i)T+(−0.5−0.866i)T2 1 + (-0.104 + 0.181i)T + (-0.5 - 0.866i)T^{2}
13 1+(0.978+1.69i)T+(−0.5+0.866i)T2 1 + (0.978 + 1.69i)T + (-0.5 + 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+(−0.669−1.15i)T+(−0.5+0.866i)T2 1 + (-0.669 - 1.15i)T + (-0.5 + 0.866i)T^{2}
29 1+(0.978−1.69i)T+(−0.5−0.866i)T2 1 + (0.978 - 1.69i)T + (-0.5 - 0.866i)T^{2}
31 1+(0.809+1.40i)T+(−0.5+0.866i)T2 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2}
41 1+(0.309+0.535i)T+(−0.5+0.866i)T2 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1−T2 1 - T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(−0.913+1.58i)T+(−0.5−0.866i)T2 1 + (-0.913 + 1.58i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.913+1.58i)T+(−0.5+0.866i)T2 1 + (0.913 + 1.58i)T + (-0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+0.209T+T2 1 + 0.209T + T^{2}
79 1+(−0.309+0.535i)T+(−0.5−0.866i)T2 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2}
83 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)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

−8.973867243513517661232743811460, −8.116431325730820265547297800561, −7.42735658366170388236890811491, −5.69123749485070751656888932915, −5.14567224547635529360986498998, −4.79851247708633225951509546469, −3.54222145757376375999772435805, −3.39886045127905311738490455685, −1.90512145991574026194646274759, −0.36592648421615334169519745721, 2.17410898921313512216720859954, 2.96277415362287566625268236106, 3.86242863986529287022263680697, 4.65421266311294312497095285270, 5.90553512470438406699376963814, 6.72982655807128968237240318451, 7.07841089127807594081823678237, 7.48082333061912539771300756200, 8.429615106178552535114094036492, 9.072361568064547331964403747495

Graph of the ZZ-function along the critical line