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Properties

Label 2-1183-7.4-c1-0-33
Degree 22
Conductor 11831183
Sign 0.989−0.145i0.989 - 0.145i
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  + (−0.545 − 0.945i)2-s + (−0.697 + 1.20i)3-s + (0.403 − 0.699i)4-s + (0.813 + 1.40i)5-s + 1.52·6-s + (2.51 + 0.817i)7-s − 3.06·8-s + (0.526 + 0.911i)9-s + (0.888 − 1.53i)10-s + (0.646 − 1.12i)11-s + (0.563 + 0.975i)12-s + (−0.600 − 2.82i)14-s − 2.26·15-s + (0.866 + 1.50i)16-s + (2.70 − 4.68i)17-s + (0.574 − 0.995i)18-s + ⋯
L(s)  = 1  + (−0.386 − 0.668i)2-s + (−0.402 + 0.697i)3-s + (0.201 − 0.349i)4-s + (0.363 + 0.629i)5-s + 0.622·6-s + (0.951 + 0.309i)7-s − 1.08·8-s + (0.175 + 0.303i)9-s + (0.280 − 0.486i)10-s + (0.195 − 0.337i)11-s + (0.162 + 0.281i)12-s + (−0.160 − 0.755i)14-s − 0.586·15-s + (0.216 + 0.375i)16-s + (0.656 − 1.13i)17-s + (0.135 − 0.234i)18-s + ⋯

Functional equation

Λ(s)=(1183s/2 ΓC(s) L(s)=((0.989−0.145i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1183s/2 ΓC(s+1/2) L(s)=((0.989−0.145i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11831183    =    7⋅1327 \cdot 13^{2}
Sign: 0.989−0.145i0.989 - 0.145i
Analytic conductor: 9.446309.44630
Root analytic conductor: 3.073483.07348
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1183(508,⋅)\chi_{1183} (508, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1183, ( :1/2), 0.989−0.145i)(2,\ 1183,\ (\ :1/2),\ 0.989 - 0.145i)

Particular Values

L(1)L(1) ≈\approx 1.4999903941.499990394
L(12)L(\frac12) ≈\approx 1.4999903941.499990394
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+(−2.51−0.817i)T 1 + (-2.51 - 0.817i)T
13 1 1
good2 1+(0.545+0.945i)T+(−1+1.73i)T2 1 + (0.545 + 0.945i)T + (-1 + 1.73i)T^{2}
3 1+(0.697−1.20i)T+(−1.5−2.59i)T2 1 + (0.697 - 1.20i)T + (-1.5 - 2.59i)T^{2}
5 1+(−0.813−1.40i)T+(−2.5+4.33i)T2 1 + (-0.813 - 1.40i)T + (-2.5 + 4.33i)T^{2}
11 1+(−0.646+1.12i)T+(−5.5−9.52i)T2 1 + (-0.646 + 1.12i)T + (-5.5 - 9.52i)T^{2}
17 1+(−2.70+4.68i)T+(−8.5−14.7i)T2 1 + (-2.70 + 4.68i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.755−1.30i)T+(−9.5+16.4i)T2 1 + (-0.755 - 1.30i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.32−2.28i)T+(−11.5+19.9i)T2 1 + (-1.32 - 2.28i)T + (-11.5 + 19.9i)T^{2}
29 1−5.81T+29T2 1 - 5.81T + 29T^{2}
31 1+(3.64−6.30i)T+(−15.5−26.8i)T2 1 + (3.64 - 6.30i)T + (-15.5 - 26.8i)T^{2}
37 1+(−3.47−6.02i)T+(−18.5+32.0i)T2 1 + (-3.47 - 6.02i)T + (-18.5 + 32.0i)T^{2}
41 1+8.09T+41T2 1 + 8.09T + 41T^{2}
43 1−11.1T+43T2 1 - 11.1T + 43T^{2}
47 1+(3.58+6.21i)T+(−23.5+40.7i)T2 1 + (3.58 + 6.21i)T + (-23.5 + 40.7i)T^{2}
53 1+(2.33−4.04i)T+(−26.5−45.8i)T2 1 + (2.33 - 4.04i)T + (-26.5 - 45.8i)T^{2}
59 1+(0.386−0.670i)T+(−29.5−51.0i)T2 1 + (0.386 - 0.670i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.37+7.57i)T+(−30.5+52.8i)T2 1 + (4.37 + 7.57i)T + (-30.5 + 52.8i)T^{2}
67 1+(3.18−5.52i)T+(−33.5−58.0i)T2 1 + (3.18 - 5.52i)T + (-33.5 - 58.0i)T^{2}
71 1−11.9T+71T2 1 - 11.9T + 71T^{2}
73 1+(3.60−6.24i)T+(−36.5−63.2i)T2 1 + (3.60 - 6.24i)T + (-36.5 - 63.2i)T^{2}
79 1+(−5.88−10.2i)T+(−39.5+68.4i)T2 1 + (-5.88 - 10.2i)T + (-39.5 + 68.4i)T^{2}
83 1−8.42T+83T2 1 - 8.42T + 83T^{2}
89 1+(−0.833−1.44i)T+(−44.5+77.0i)T2 1 + (-0.833 - 1.44i)T + (-44.5 + 77.0i)T^{2}
97 1−12.4T+97T2 1 - 12.4T + 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

−10.03107094629136001744456225814, −9.294303090096873402054408188280, −8.375066576898917722760102717780, −7.31062067112113098128613304037, −6.31994161506721318559623967870, −5.37993339973484056490471316796, −4.82099774790752135166678678067, −3.34645791456553755334384610974, −2.38550978657888213863865991480, −1.21960444227231288825978046579, 0.932335810757812296275792347703, 2.05090897808161313620828567569, 3.65436556821323127284734939426, 4.77065940932360376720580795879, 5.85076779960809742061798044951, 6.45454828655831768098757136480, 7.46360647993130181732000691002, 7.83100842311482537112418372902, 8.808189712712310022052337669856, 9.421243489426143895251855507725

Graph of the ZZ-function along the critical line