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Properties

Label 2-369-9.4-c1-0-9
Degree 22
Conductor 369369
Sign 0.999+0.0252i0.999 + 0.0252i
Analytic cond. 2.946472.94647
Root an. cond. 1.716531.71653
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  + (−0.395 − 0.684i)2-s + (−1.70 + 0.279i)3-s + (0.687 − 1.19i)4-s + (−0.0693 + 0.120i)5-s + (0.866 + 1.05i)6-s + (2.32 + 4.02i)7-s − 2.66·8-s + (2.84 − 0.954i)9-s + 0.109·10-s + (−0.684 − 1.18i)11-s + (−0.842 + 2.22i)12-s + (−3.08 + 5.33i)13-s + (1.83 − 3.18i)14-s + (0.0850 − 0.224i)15-s + (−0.321 − 0.556i)16-s + 6.39·17-s + ⋯
L(s)  = 1  + (−0.279 − 0.484i)2-s + (−0.986 + 0.161i)3-s + (0.343 − 0.595i)4-s + (−0.0310 + 0.0537i)5-s + (0.353 + 0.432i)6-s + (0.878 + 1.52i)7-s − 0.943·8-s + (0.948 − 0.318i)9-s + 0.0346·10-s + (−0.206 − 0.357i)11-s + (−0.243 + 0.643i)12-s + (−0.854 + 1.48i)13-s + (0.491 − 0.850i)14-s + (0.0219 − 0.0580i)15-s + (−0.0802 − 0.139i)16-s + 1.55·17-s + ⋯

Functional equation

Λ(s)=(369s/2 ΓC(s) L(s)=((0.999+0.0252i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0252i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(369s/2 ΓC(s+1/2) L(s)=((0.999+0.0252i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0252i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 369369    =    32⋅413^{2} \cdot 41
Sign: 0.999+0.0252i0.999 + 0.0252i
Analytic conductor: 2.946472.94647
Root analytic conductor: 1.716531.71653
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ369(247,⋅)\chi_{369} (247, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 369, ( :1/2), 0.999+0.0252i)(2,\ 369,\ (\ :1/2),\ 0.999 + 0.0252i)

Particular Values

L(1)L(1) ≈\approx 0.986268−0.0124548i0.986268 - 0.0124548i
L(12)L(\frac12) ≈\approx 0.986268−0.0124548i0.986268 - 0.0124548i
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+(1.70−0.279i)T 1 + (1.70 - 0.279i)T
41 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good2 1+(0.395+0.684i)T+(−1+1.73i)T2 1 + (0.395 + 0.684i)T + (-1 + 1.73i)T^{2}
5 1+(0.0693−0.120i)T+(−2.5−4.33i)T2 1 + (0.0693 - 0.120i)T + (-2.5 - 4.33i)T^{2}
7 1+(−2.32−4.02i)T+(−3.5+6.06i)T2 1 + (-2.32 - 4.02i)T + (-3.5 + 6.06i)T^{2}
11 1+(0.684+1.18i)T+(−5.5+9.52i)T2 1 + (0.684 + 1.18i)T + (-5.5 + 9.52i)T^{2}
13 1+(3.08−5.33i)T+(−6.5−11.2i)T2 1 + (3.08 - 5.33i)T + (-6.5 - 11.2i)T^{2}
17 1−6.39T+17T2 1 - 6.39T + 17T^{2}
19 1−3.98T+19T2 1 - 3.98T + 19T^{2}
23 1+(−2.79+4.84i)T+(−11.5−19.9i)T2 1 + (-2.79 + 4.84i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.62−2.80i)T+(−14.5+25.1i)T2 1 + (-1.62 - 2.80i)T + (-14.5 + 25.1i)T^{2}
31 1+(−3.67+6.36i)T+(−15.5−26.8i)T2 1 + (-3.67 + 6.36i)T + (-15.5 - 26.8i)T^{2}
37 1−3.00T+37T2 1 - 3.00T + 37T^{2}
43 1+(0.911+1.57i)T+(−21.5+37.2i)T2 1 + (0.911 + 1.57i)T + (-21.5 + 37.2i)T^{2}
47 1+(−3.89−6.75i)T+(−23.5+40.7i)T2 1 + (-3.89 - 6.75i)T + (-23.5 + 40.7i)T^{2}
53 1+8.89T+53T2 1 + 8.89T + 53T^{2}
59 1+(0.751−1.30i)T+(−29.5−51.0i)T2 1 + (0.751 - 1.30i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.283−0.490i)T+(−30.5+52.8i)T2 1 + (-0.283 - 0.490i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.62+2.82i)T+(−33.5−58.0i)T2 1 + (-1.62 + 2.82i)T + (-33.5 - 58.0i)T^{2}
71 1+9.51T+71T2 1 + 9.51T + 71T^{2}
73 1+2.79T+73T2 1 + 2.79T + 73T^{2}
79 1+(−1.47−2.54i)T+(−39.5+68.4i)T2 1 + (-1.47 - 2.54i)T + (-39.5 + 68.4i)T^{2}
83 1+(4.39+7.61i)T+(−41.5+71.8i)T2 1 + (4.39 + 7.61i)T + (-41.5 + 71.8i)T^{2}
89 1−0.173T+89T2 1 - 0.173T + 89T^{2}
97 1+(−4.90−8.48i)T+(−48.5+84.0i)T2 1 + (-4.90 - 8.48i)T + (-48.5 + 84.0i)T^{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

−11.58939777827283803400400683639, −10.66214189652723325528586755774, −9.619796861391115430117152189987, −9.042005737712611233543562062157, −7.58096773040552827039645891916, −6.36011873233499369247159709933, −5.50050964446694268776183657858, −4.80826858211516856628153316739, −2.74207876006419458607058220340, −1.39305856144094617386071280861, 0.979973838076168965257486311897, 3.23147349857186246475388103978, 4.70108437411828508678452760102, 5.58404449524613548911368507523, 6.97003920467333678910625126717, 7.60890157296133683556928983596, 8.021204866697926934345574326282, 9.915105491305756740978488175178, 10.42998005010448271400764999733, 11.46354359755764195721006083268

Graph of the ZZ-function along the critical line