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Properties

Label 2-153-17.9-c3-0-7
Degree 22
Conductor 153153
Sign −0.256−0.966i-0.256 - 0.966i
Analytic cond. 9.027299.02729
Root an. cond. 3.004543.00454
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.320 + 0.320i)2-s + 7.79i·4-s + (2.81 − 6.79i)5-s + (6.62 + 15.9i)7-s + (−5.05 − 5.05i)8-s + (1.27 + 3.07i)10-s + (0.780 − 0.323i)11-s + 29.3i·13-s + (−7.23 − 2.99i)14-s − 59.1·16-s + (14.8 + 68.5i)17-s + (−13.1 + 13.1i)19-s + (52.9 + 21.9i)20-s + (−0.146 + 0.353i)22-s + (−111. + 46.0i)23-s + ⋯
L(s)  = 1  + (−0.113 + 0.113i)2-s + 0.974i·4-s + (0.251 − 0.607i)5-s + (0.357 + 0.863i)7-s + (−0.223 − 0.223i)8-s + (0.0403 + 0.0972i)10-s + (0.0213 − 0.00885i)11-s + 0.626i·13-s + (−0.138 − 0.0572i)14-s − 0.923·16-s + (0.211 + 0.977i)17-s + (−0.158 + 0.158i)19-s + (0.592 + 0.245i)20-s + (−0.00141 + 0.00342i)22-s + (−1.00 + 0.417i)23-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=((−0.256−0.966i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+3/2) L(s)=((−0.256−0.966i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: −0.256−0.966i-0.256 - 0.966i
Analytic conductor: 9.027299.02729
Root analytic conductor: 3.004543.00454
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ153(145,⋅)\chi_{153} (145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 153, ( :3/2), −0.256−0.966i)(2,\ 153,\ (\ :3/2),\ -0.256 - 0.966i)

Particular Values

L(2)L(2) ≈\approx 0.869812+1.13127i0.869812 + 1.13127i
L(12)L(\frac12) ≈\approx 0.869812+1.13127i0.869812 + 1.13127i
L(52)L(\frac{5}{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
17 1+(−14.8−68.5i)T 1 + (-14.8 - 68.5i)T
good2 1+(0.320−0.320i)T−8iT2 1 + (0.320 - 0.320i)T - 8iT^{2}
5 1+(−2.81+6.79i)T+(−88.3−88.3i)T2 1 + (-2.81 + 6.79i)T + (-88.3 - 88.3i)T^{2}
7 1+(−6.62−15.9i)T+(−242.+242.i)T2 1 + (-6.62 - 15.9i)T + (-242. + 242. i)T^{2}
11 1+(−0.780+0.323i)T+(941.−941.i)T2 1 + (-0.780 + 0.323i)T + (941. - 941. i)T^{2}
13 1−29.3iT−2.19e3T2 1 - 29.3iT - 2.19e3T^{2}
19 1+(13.1−13.1i)T−6.85e3iT2 1 + (13.1 - 13.1i)T - 6.85e3iT^{2}
23 1+(111.−46.0i)T+(8.60e3−8.60e3i)T2 1 + (111. - 46.0i)T + (8.60e3 - 8.60e3i)T^{2}
29 1+(76.9−185.i)T+(−1.72e4−1.72e4i)T2 1 + (76.9 - 185. i)T + (-1.72e4 - 1.72e4i)T^{2}
31 1+(−98.5−40.8i)T+(2.10e4+2.10e4i)T2 1 + (-98.5 - 40.8i)T + (2.10e4 + 2.10e4i)T^{2}
37 1+(64.8+26.8i)T+(3.58e4+3.58e4i)T2 1 + (64.8 + 26.8i)T + (3.58e4 + 3.58e4i)T^{2}
41 1+(73.4+177.i)T+(−4.87e4+4.87e4i)T2 1 + (73.4 + 177. i)T + (-4.87e4 + 4.87e4i)T^{2}
43 1+(−359.−359.i)T+7.95e4iT2 1 + (-359. - 359. i)T + 7.95e4iT^{2}
47 1+235.iT−1.03e5T2 1 + 235. iT - 1.03e5T^{2}
53 1+(−36.0+36.0i)T−1.48e5iT2 1 + (-36.0 + 36.0i)T - 1.48e5iT^{2}
59 1+(106.+106.i)T+2.05e5iT2 1 + (106. + 106. i)T + 2.05e5iT^{2}
61 1+(355.+857.i)T+(−1.60e5+1.60e5i)T2 1 + (355. + 857. i)T + (-1.60e5 + 1.60e5i)T^{2}
67 1−158.T+3.00e5T2 1 - 158.T + 3.00e5T^{2}
71 1+(−536.−222.i)T+(2.53e5+2.53e5i)T2 1 + (-536. - 222. i)T + (2.53e5 + 2.53e5i)T^{2}
73 1+(−186.+449.i)T+(−2.75e5−2.75e5i)T2 1 + (-186. + 449. i)T + (-2.75e5 - 2.75e5i)T^{2}
79 1+(−1.02e3+425.i)T+(3.48e5−3.48e5i)T2 1 + (-1.02e3 + 425. i)T + (3.48e5 - 3.48e5i)T^{2}
83 1+(−1.03e3+1.03e3i)T−5.71e5iT2 1 + (-1.03e3 + 1.03e3i)T - 5.71e5iT^{2}
89 1−1.04e3iT−7.04e5T2 1 - 1.04e3iT - 7.04e5T^{2}
97 1+(195.−473.i)T+(−6.45e5−6.45e5i)T2 1 + (195. - 473. i)T + (-6.45e5 - 6.45e5i)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

−12.54791668679482442441369123405, −12.11319755347895922517323388641, −10.93788168303943406857767352817, −9.384147806884856907817501491562, −8.663306535570823758203376571460, −7.77931031819678472184006285345, −6.37280215354431856232718180577, −5.05568252820361594017861009470, −3.66015506213464349738784065603, −1.94708982368249383095366849493, 0.71937640971803849520897218072, 2.48327013777258260344406148047, 4.35899709980407091685709922895, 5.69589381794241845904240362942, 6.78200447609114429878267908391, 7.937005338184842280296721127564, 9.434242451142644248645391316593, 10.32782285260299049917756563100, 10.90156904884865161989783171430, 12.02985562927873270454337056644

Graph of the ZZ-function along the critical line