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Properties

Label 2-1183-7.2-c1-0-39
Degree 22
Conductor 11831183
Sign 0.203+0.979i0.203 + 0.979i
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  + (1.27 − 2.20i)2-s + (−0.156 − 0.270i)3-s + (−2.22 − 3.85i)4-s + (−1.54 + 2.66i)5-s − 0.795·6-s + (1.63 + 2.08i)7-s − 6.24·8-s + (1.45 − 2.51i)9-s + (3.91 + 6.78i)10-s + (2.10 + 3.64i)11-s + (−0.697 + 1.20i)12-s + (6.65 − 0.946i)14-s + 0.964·15-s + (−3.47 + 6.01i)16-s + (2.98 + 5.16i)17-s + (−3.68 − 6.38i)18-s + ⋯
L(s)  = 1  + (0.898 − 1.55i)2-s + (−0.0903 − 0.156i)3-s + (−1.11 − 1.92i)4-s + (−0.689 + 1.19i)5-s − 0.324·6-s + (0.616 + 0.786i)7-s − 2.20·8-s + (0.483 − 0.837i)9-s + (1.23 + 2.14i)10-s + (0.634 + 1.09i)11-s + (−0.201 + 0.348i)12-s + (1.77 − 0.253i)14-s + 0.248·15-s + (−0.868 + 1.50i)16-s + (0.723 + 1.25i)17-s + (−0.869 − 1.50i)18-s + ⋯

Functional equation

Λ(s)=(1183s/2 ΓC(s) L(s)=((0.203+0.979i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1183s/2 ΓC(s+1/2) L(s)=((0.203+0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11831183    =    7⋅1327 \cdot 13^{2}
Sign: 0.203+0.979i0.203 + 0.979i
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), 0.203+0.979i)(2,\ 1183,\ (\ :1/2),\ 0.203 + 0.979i)

Particular Values

L(1)L(1) ≈\approx 2.4411848672.441184867
L(12)L(\frac12) ≈\approx 2.4411848672.441184867
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+(−1.63−2.08i)T 1 + (-1.63 - 2.08i)T
13 1 1
good2 1+(−1.27+2.20i)T+(−1−1.73i)T2 1 + (-1.27 + 2.20i)T + (-1 - 1.73i)T^{2}
3 1+(0.156+0.270i)T+(−1.5+2.59i)T2 1 + (0.156 + 0.270i)T + (-1.5 + 2.59i)T^{2}
5 1+(1.54−2.66i)T+(−2.5−4.33i)T2 1 + (1.54 - 2.66i)T + (-2.5 - 4.33i)T^{2}
11 1+(−2.10−3.64i)T+(−5.5+9.52i)T2 1 + (-2.10 - 3.64i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2.98−5.16i)T+(−8.5+14.7i)T2 1 + (-2.98 - 5.16i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.50+2.61i)T+(−9.5−16.4i)T2 1 + (-1.50 + 2.61i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.86+4.96i)T+(−11.5−19.9i)T2 1 + (-2.86 + 4.96i)T + (-11.5 - 19.9i)T^{2}
29 1−1.74T+29T2 1 - 1.74T + 29T^{2}
31 1+(−0.336−0.583i)T+(−15.5+26.8i)T2 1 + (-0.336 - 0.583i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.539−0.935i)T+(−18.5−32.0i)T2 1 + (0.539 - 0.935i)T + (-18.5 - 32.0i)T^{2}
41 1−8.51T+41T2 1 - 8.51T + 41T^{2}
43 1+1.23T+43T2 1 + 1.23T + 43T^{2}
47 1+(5.51−9.55i)T+(−23.5−40.7i)T2 1 + (5.51 - 9.55i)T + (-23.5 - 40.7i)T^{2}
53 1+(1.87+3.24i)T+(−26.5+45.8i)T2 1 + (1.87 + 3.24i)T + (-26.5 + 45.8i)T^{2}
59 1+(2.08+3.60i)T+(−29.5+51.0i)T2 1 + (2.08 + 3.60i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.50+2.61i)T+(−30.5−52.8i)T2 1 + (-1.50 + 2.61i)T + (-30.5 - 52.8i)T^{2}
67 1+(−4.62−8.00i)T+(−33.5+58.0i)T2 1 + (-4.62 - 8.00i)T + (-33.5 + 58.0i)T^{2}
71 1−9.19T+71T2 1 - 9.19T + 71T^{2}
73 1+(−2.17−3.77i)T+(−36.5+63.2i)T2 1 + (-2.17 - 3.77i)T + (-36.5 + 63.2i)T^{2}
79 1+(−2.33+4.03i)T+(−39.5−68.4i)T2 1 + (-2.33 + 4.03i)T + (-39.5 - 68.4i)T^{2}
83 1−13.0T+83T2 1 - 13.0T + 83T^{2}
89 1+(−6.87+11.9i)T+(−44.5−77.0i)T2 1 + (-6.87 + 11.9i)T + (-44.5 - 77.0i)T^{2}
97 1+12.8T+97T2 1 + 12.8T + 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

−9.861246135826927248882361718201, −9.205187279315495932967451165240, −7.967619857779899500668805147013, −6.86065444360385007950873015500, −6.13186274099148917473295853789, −4.89305527162285390862693307897, −4.11533493327683035432610783307, −3.30544129031472545279327246776, −2.39005051324098278917139504739, −1.28206162255038493908372412817, 1.01304595279441183184576138250, 3.50846469753943257383778528657, 4.16582371251339608535247193755, 5.09356598777295933037047571602, 5.33848841664199952442800996397, 6.64096157532265902847195143069, 7.71734671656894112297326500097, 7.79204363544474646877021745908, 8.725123538379872262559146302456, 9.588562256510929063810900768503

Graph of the ZZ-function along the critical line