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Properties

Label 2-1183-7.2-c1-0-20
Degree 22
Conductor 11831183
Sign −0.605+0.795i-0.605 + 0.795i
Analytic cond. 9.446309.44630
Root an. cond. 3.073483.07348
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  + (−0.5 + 0.866i)2-s + (1.5 + 2.59i)3-s + (0.500 + 0.866i)4-s + (−1.5 + 2.59i)5-s − 3·6-s + (0.5 − 2.59i)7-s − 3·8-s + (−3 + 5.19i)9-s + (−1.5 − 2.59i)10-s + (1.5 + 2.59i)11-s + (−1.49 + 2.59i)12-s + (2 + 1.73i)14-s − 9·15-s + (0.500 − 0.866i)16-s + (1 + 1.73i)17-s + (−3 − 5.19i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.866 + 1.49i)3-s + (0.250 + 0.433i)4-s + (−0.670 + 1.16i)5-s − 1.22·6-s + (0.188 − 0.981i)7-s − 1.06·8-s + (−1 + 1.73i)9-s + (−0.474 − 0.821i)10-s + (0.452 + 0.783i)11-s + (−0.433 + 0.749i)12-s + (0.534 + 0.462i)14-s − 2.32·15-s + (0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s + (−0.707 − 1.22i)18-s + ⋯

Functional equation

Λ(s)=(1183s/2 ΓC(s) L(s)=((−0.605+0.795i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1183s/2 ΓC(s+1/2) L(s)=((−0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11831183    =    7⋅1327 \cdot 13^{2}
Sign: −0.605+0.795i-0.605 + 0.795i
Analytic conductor: 9.446309.44630
Root analytic conductor: 3.073483.07348
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1183(170,⋅)\chi_{1183} (170, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1183, ( :1/2), −0.605+0.795i)(2,\ 1183,\ (\ :1/2),\ -0.605 + 0.795i)

Particular Values

L(1)L(1) ≈\approx 1.5388555971.538855597
L(12)L(\frac12) ≈\approx 1.5388555971.538855597
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)
bad7 1+(−0.5+2.59i)T 1 + (-0.5 + 2.59i)T
13 1 1
good2 1+(0.5−0.866i)T+(−1−1.73i)T2 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2}
3 1+(−1.5−2.59i)T+(−1.5+2.59i)T2 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2}
5 1+(1.5−2.59i)T+(−2.5−4.33i)T2 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2}
11 1+(−1.5−2.59i)T+(−5.5+9.52i)T2 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1−1.73i)T+(−8.5+14.7i)T2 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.5+0.866i)T+(−9.5−16.4i)T2 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1−7T+29T2 1 - 7T + 29T^{2}
31 1+(1.5+2.59i)T+(−15.5+26.8i)T2 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2}
37 1+(1−1.73i)T+(−18.5−32.0i)T2 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2}
41 1−3T+41T2 1 - 3T + 41T^{2}
43 1+7T+43T2 1 + 7T + 43T^{2}
47 1+(0.5−0.866i)T+(−23.5−40.7i)T2 1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2}
53 1+(1.5+2.59i)T+(−26.5+45.8i)T2 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2}
59 1+(−2−3.46i)T+(−29.5+51.0i)T2 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2}
61 1+(−6.5+11.2i)T+(−30.5−52.8i)T2 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.5−2.59i)T+(−33.5+58.0i)T2 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2}
71 1−13T+71T2 1 - 13T + 71T^{2}
73 1+(−6.5−11.2i)T+(−36.5+63.2i)T2 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2}
79 1+(−1.5+2.59i)T+(−39.5−68.4i)T2 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(3−5.19i)T+(−44.5−77.0i)T2 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2}
97 1+5T+97T2 1 + 5T + 97T^{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

−10.06079988770602998195650327371, −9.589472085088987738886881322831, −8.457303233765370896824918830598, −7.960343463594599175013528876389, −7.13384198679466693510945879505, −6.51996515340616316203865059603, −4.92776849126758214528802585870, −3.90684296154350397270770949835, −3.50523013181163719205874532464, −2.51767491301095379565245519379, 0.68640472950616425433565807329, 1.50910939711207774995178201881, 2.53984078477029957018061134867, 3.44397887691165123366912573880, 5.06524301912545564168067974061, 6.01276094989407434318296635795, 6.81035681456118698446561915609, 7.924630696192998327259565704011, 8.601413937958611170742838679339, 8.880551276128510091210319388905

Graph of the ZZ-function along the critical line