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Properties

Label 2-1183-7.2-c1-0-56
Degree 22
Conductor 11831183
Sign 0.443+0.896i0.443 + 0.896i
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  + (0.777 − 1.34i)2-s + (0.244 + 0.423i)3-s + (−0.208 − 0.361i)4-s + (−0.595 + 1.03i)5-s + 0.760·6-s + (−2.10 − 1.60i)7-s + 2.46·8-s + (1.38 − 2.39i)9-s + (0.926 + 1.60i)10-s + (1.05 + 1.83i)11-s + (0.102 − 0.176i)12-s + (−3.79 + 1.58i)14-s − 0.582·15-s + (2.33 − 4.03i)16-s + (0.453 + 0.784i)17-s + (−2.14 − 3.71i)18-s + ⋯
L(s)  = 1  + (0.549 − 0.952i)2-s + (0.141 + 0.244i)3-s + (−0.104 − 0.180i)4-s + (−0.266 + 0.461i)5-s + 0.310·6-s + (−0.795 − 0.606i)7-s + 0.870·8-s + (0.460 − 0.796i)9-s + (0.292 + 0.507i)10-s + (0.319 + 0.552i)11-s + (0.0294 − 0.0510i)12-s + (−1.01 + 0.423i)14-s − 0.150·15-s + (0.582 − 1.00i)16-s + (0.109 + 0.190i)17-s + (−0.505 − 0.876i)18-s + ⋯

Functional equation

Λ(s)=(1183s/2 ΓC(s) L(s)=((0.443+0.896i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1183s/2 ΓC(s+1/2) L(s)=((0.443+0.896i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11831183    =    7⋅1327 \cdot 13^{2}
Sign: 0.443+0.896i0.443 + 0.896i
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), 0.443+0.896i)(2,\ 1183,\ (\ :1/2),\ 0.443 + 0.896i)

Particular Values

L(1)L(1) ≈\approx 2.4226795552.422679555
L(12)L(\frac12) ≈\approx 2.4226795552.422679555
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+(2.10+1.60i)T 1 + (2.10 + 1.60i)T
13 1 1
good2 1+(−0.777+1.34i)T+(−1−1.73i)T2 1 + (-0.777 + 1.34i)T + (-1 - 1.73i)T^{2}
3 1+(−0.244−0.423i)T+(−1.5+2.59i)T2 1 + (-0.244 - 0.423i)T + (-1.5 + 2.59i)T^{2}
5 1+(0.595−1.03i)T+(−2.5−4.33i)T2 1 + (0.595 - 1.03i)T + (-2.5 - 4.33i)T^{2}
11 1+(−1.05−1.83i)T+(−5.5+9.52i)T2 1 + (-1.05 - 1.83i)T + (-5.5 + 9.52i)T^{2}
17 1+(−0.453−0.784i)T+(−8.5+14.7i)T2 1 + (-0.453 - 0.784i)T + (-8.5 + 14.7i)T^{2}
19 1+(−3.34+5.79i)T+(−9.5−16.4i)T2 1 + (-3.34 + 5.79i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.79−3.11i)T+(−11.5−19.9i)T2 1 + (1.79 - 3.11i)T + (-11.5 - 19.9i)T^{2}
29 1−8.51T+29T2 1 - 8.51T + 29T^{2}
31 1+(2.64+4.57i)T+(−15.5+26.8i)T2 1 + (2.64 + 4.57i)T + (-15.5 + 26.8i)T^{2}
37 1+(−2.49+4.32i)T+(−18.5−32.0i)T2 1 + (-2.49 + 4.32i)T + (-18.5 - 32.0i)T^{2}
41 1+1.53T+41T2 1 + 1.53T + 41T^{2}
43 1−5.43T+43T2 1 - 5.43T + 43T^{2}
47 1+(1.59−2.75i)T+(−23.5−40.7i)T2 1 + (1.59 - 2.75i)T + (-23.5 - 40.7i)T^{2}
53 1+(−1.41−2.44i)T+(−26.5+45.8i)T2 1 + (-1.41 - 2.44i)T + (-26.5 + 45.8i)T^{2}
59 1+(5.12+8.87i)T+(−29.5+51.0i)T2 1 + (5.12 + 8.87i)T + (-29.5 + 51.0i)T^{2}
61 1+(−4.13+7.16i)T+(−30.5−52.8i)T2 1 + (-4.13 + 7.16i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.87+3.24i)T+(−33.5+58.0i)T2 1 + (1.87 + 3.24i)T + (-33.5 + 58.0i)T^{2}
71 1−2.53T+71T2 1 - 2.53T + 71T^{2}
73 1+(2.86+4.96i)T+(−36.5+63.2i)T2 1 + (2.86 + 4.96i)T + (-36.5 + 63.2i)T^{2}
79 1+(3.03−5.25i)T+(−39.5−68.4i)T2 1 + (3.03 - 5.25i)T + (-39.5 - 68.4i)T^{2}
83 1−11.6T+83T2 1 - 11.6T + 83T^{2}
89 1+(8.87−15.3i)T+(−44.5−77.0i)T2 1 + (8.87 - 15.3i)T + (-44.5 - 77.0i)T^{2}
97 1+6.20T+97T2 1 + 6.20T + 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

−9.663462627797668557186692487970, −9.313622942302346456573153905952, −7.77330203229884603687212143098, −7.10097167260249919661625884375, −6.43347808451808237585485567288, −4.94446309557233005908152603539, −4.02561259918712677504957173439, −3.45405627960306554778699978084, −2.60294050096516126781072825289, −1.04924157333087649608616258112, 1.31419221073868059339927177403, 2.78485035540348492701607993728, 4.06535255803155628758156935480, 4.95525213408729393763373886049, 5.78932040341223639429079743594, 6.49030320296202459605901133032, 7.30213037947084382964322267532, 8.188326779350736915381404210398, 8.722159991304477519653849260292, 10.03812672363387026260317802677

Graph of the ZZ-function along the critical line