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Properties

Label 2-147-7.2-c1-0-3
Degree 22
Conductor 147147
Sign 0.749−0.661i0.749 - 0.661i
Analytic cond. 1.173801.17380
Root an. cond. 1.083421.08342
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  + (−0.207 + 0.358i)2-s + (0.5 + 0.866i)3-s + (0.914 + 1.58i)4-s + (1.70 − 2.95i)5-s − 0.414·6-s − 1.58·8-s + (−0.499 + 0.866i)9-s + (0.707 + 1.22i)10-s + (1 + 1.73i)11-s + (−0.914 + 1.58i)12-s − 2.58·13-s + 3.41·15-s + (−1.49 + 2.59i)16-s + (−1.12 − 1.94i)17-s + (−0.207 − 0.358i)18-s + (−1.41 + 2.44i)19-s + ⋯
L(s)  = 1  + (−0.146 + 0.253i)2-s + (0.288 + 0.499i)3-s + (0.457 + 0.791i)4-s + (0.763 − 1.32i)5-s − 0.169·6-s − 0.560·8-s + (−0.166 + 0.288i)9-s + (0.223 + 0.387i)10-s + (0.301 + 0.522i)11-s + (−0.263 + 0.457i)12-s − 0.717·13-s + 0.881·15-s + (−0.374 + 0.649i)16-s + (−0.271 − 0.471i)17-s + (−0.0488 − 0.0845i)18-s + (−0.324 + 0.561i)19-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((0.749−0.661i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+1/2) L(s)=((0.749−0.661i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: 0.749−0.661i0.749 - 0.661i
Analytic conductor: 1.173801.17380
Root analytic conductor: 1.083421.08342
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ147(79,⋅)\chi_{147} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :1/2), 0.749−0.661i)(2,\ 147,\ (\ :1/2),\ 0.749 - 0.661i)

Particular Values

L(1)L(1) ≈\approx 1.21112+0.457928i1.21112 + 0.457928i
L(12)L(\frac12) ≈\approx 1.21112+0.457928i1.21112 + 0.457928i
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)
bad3 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1 1
good2 1+(0.207−0.358i)T+(−1−1.73i)T2 1 + (0.207 - 0.358i)T + (-1 - 1.73i)T^{2}
5 1+(−1.70+2.95i)T+(−2.5−4.33i)T2 1 + (-1.70 + 2.95i)T + (-2.5 - 4.33i)T^{2}
11 1+(−1−1.73i)T+(−5.5+9.52i)T2 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2}
13 1+2.58T+13T2 1 + 2.58T + 13T^{2}
17 1+(1.12+1.94i)T+(−8.5+14.7i)T2 1 + (1.12 + 1.94i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.41−2.44i)T+(−9.5−16.4i)T2 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.82+6.63i)T+(−11.5−19.9i)T2 1 + (-3.82 + 6.63i)T + (-11.5 - 19.9i)T^{2}
29 1+6.82T+29T2 1 + 6.82T + 29T^{2}
31 1+(0.585+1.01i)T+(−15.5+26.8i)T2 1 + (0.585 + 1.01i)T + (-15.5 + 26.8i)T^{2}
37 1+(−2+3.46i)T+(−18.5−32.0i)T2 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2}
41 1+6.24T+41T2 1 + 6.24T + 41T^{2}
43 1−5.65T+43T2 1 - 5.65T + 43T^{2}
47 1+(1.41−2.44i)T+(−23.5−40.7i)T2 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2}
53 1+(−1−1.73i)T+(−26.5+45.8i)T2 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2}
59 1+(0.585+1.01i)T+(−29.5+51.0i)T2 1 + (0.585 + 1.01i)T + (-29.5 + 51.0i)T^{2}
61 1+(−6.12+10.6i)T+(−30.5−52.8i)T2 1 + (-6.12 + 10.6i)T + (-30.5 - 52.8i)T^{2}
67 1+(−2.82−4.89i)T+(−33.5+58.0i)T2 1 + (-2.82 - 4.89i)T + (-33.5 + 58.0i)T^{2}
71 1−9.31T+71T2 1 - 9.31T + 71T^{2}
73 1+(−6.94−12.0i)T+(−36.5+63.2i)T2 1 + (-6.94 - 12.0i)T + (-36.5 + 63.2i)T^{2}
79 1+(6.82−11.8i)T+(−39.5−68.4i)T2 1 + (6.82 - 11.8i)T + (-39.5 - 68.4i)T^{2}
83 1+7.31T+83T2 1 + 7.31T + 83T^{2}
89 1+(7.12−12.3i)T+(−44.5−77.0i)T2 1 + (7.12 - 12.3i)T + (-44.5 - 77.0i)T^{2}
97 1+2.58T+97T2 1 + 2.58T + 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

−12.84605580745188520116095561807, −12.52093737635660074777789998122, −11.20430087055643746324755995282, −9.762587299902956233208971903441, −9.035351245279866829599741745252, −8.136709774759497825112418332544, −6.84111233223846220194210314734, −5.35204718184660538430895470686, −4.17526723950096684770236833405, −2.30619245105572922245043296140, 1.93060194382655682929945983018, 3.11633590144782128325846682748, 5.56402701137154873391531483432, 6.55030564562380442777133054919, 7.32111794603211418314867556170, 9.078299705054228347854104539727, 9.976648794703119148991022507653, 10.93508584965005132476859004473, 11.62050185568625122740774259941, 13.14047675504863346171648618576

Graph of the ZZ-function along the critical line