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Properties

Label 2-152-152.3-c1-0-7
Degree 22
Conductor 152152
Sign 0.973+0.228i0.973 + 0.228i
Analytic cond. 1.213721.21372
Root an. cond. 1.101691.10169
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  + (−0.909 + 1.08i)2-s + (−0.301 − 0.827i)3-s + (−0.347 − 1.96i)4-s + (1.17 + 0.426i)6-s + (2.44 + 1.41i)8-s + (1.70 − 1.42i)9-s + (3.30 − 5.72i)11-s + (−1.52 + 0.881i)12-s + (−3.75 + 1.36i)16-s + (6.05 + 5.07i)17-s + 3.14i·18-s + (−4.35 − 0.248i)19-s + (3.19 + 8.77i)22-s + (0.432 − 2.45i)24-s + (−4.69 − 1.71i)25-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.173 − 0.478i)3-s + (−0.173 − 0.984i)4-s + (0.478 + 0.173i)6-s + (0.866 + 0.500i)8-s + (0.567 − 0.476i)9-s + (0.995 − 1.72i)11-s + (−0.440 + 0.254i)12-s + (−0.939 + 0.342i)16-s + (1.46 + 1.23i)17-s + 0.741i·18-s + (−0.998 − 0.0569i)19-s + (0.681 + 1.87i)22-s + (0.0883 − 0.500i)24-s + (−0.939 − 0.342i)25-s + ⋯

Functional equation

Λ(s)=(152s/2 ΓC(s) L(s)=((0.973+0.228i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(152s/2 ΓC(s+1/2) L(s)=((0.973+0.228i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 152152    =    23⋅192^{3} \cdot 19
Sign: 0.973+0.228i0.973 + 0.228i
Analytic conductor: 1.213721.21372
Root analytic conductor: 1.101691.10169
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ152(3,⋅)\chi_{152} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 152, ( :1/2), 0.973+0.228i)(2,\ 152,\ (\ :1/2),\ 0.973 + 0.228i)

Particular Values

L(1)L(1) ≈\approx 0.813199−0.0942721i0.813199 - 0.0942721i
L(12)L(\frac12) ≈\approx 0.813199−0.0942721i0.813199 - 0.0942721i
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)
bad2 1+(0.909−1.08i)T 1 + (0.909 - 1.08i)T
19 1+(4.35+0.248i)T 1 + (4.35 + 0.248i)T
good3 1+(0.301+0.827i)T+(−2.29+1.92i)T2 1 + (0.301 + 0.827i)T + (-2.29 + 1.92i)T^{2}
5 1+(4.69+1.71i)T2 1 + (4.69 + 1.71i)T^{2}
7 1+(3.5−6.06i)T2 1 + (3.5 - 6.06i)T^{2}
11 1+(−3.30+5.72i)T+(−5.5−9.52i)T2 1 + (-3.30 + 5.72i)T + (-5.5 - 9.52i)T^{2}
13 1+(9.95+8.35i)T2 1 + (9.95 + 8.35i)T^{2}
17 1+(−6.05−5.07i)T+(2.95+16.7i)T2 1 + (-6.05 - 5.07i)T + (2.95 + 16.7i)T^{2}
23 1+(21.6−7.86i)T2 1 + (21.6 - 7.86i)T^{2}
29 1+(5.03−28.5i)T2 1 + (5.03 - 28.5i)T^{2}
31 1+(−15.5+26.8i)T2 1 + (-15.5 + 26.8i)T^{2}
37 1+37T2 1 + 37T^{2}
41 1+(−2.69−7.39i)T+(−31.4+26.3i)T2 1 + (-2.69 - 7.39i)T + (-31.4 + 26.3i)T^{2}
43 1+(−0.407+2.31i)T+(−40.4−14.7i)T2 1 + (-0.407 + 2.31i)T + (-40.4 - 14.7i)T^{2}
47 1+(−8.16+46.2i)T2 1 + (-8.16 + 46.2i)T^{2}
53 1+(−49.8+18.1i)T2 1 + (-49.8 + 18.1i)T^{2}
59 1+(7.22−8.60i)T+(−10.2−58.1i)T2 1 + (7.22 - 8.60i)T + (-10.2 - 58.1i)T^{2}
61 1+(57.3−20.8i)T2 1 + (57.3 - 20.8i)T^{2}
67 1+(−9.80−11.6i)T+(−11.6+65.9i)T2 1 + (-9.80 - 11.6i)T + (-11.6 + 65.9i)T^{2}
71 1+(−66.7−24.2i)T2 1 + (-66.7 - 24.2i)T^{2}
73 1+(−3.68+1.34i)T+(55.9−46.9i)T2 1 + (-3.68 + 1.34i)T + (55.9 - 46.9i)T^{2}
79 1+(60.5−50.7i)T2 1 + (60.5 - 50.7i)T^{2}
83 1+(−2.95−5.11i)T+(−41.5+71.8i)T2 1 + (-2.95 - 5.11i)T + (-41.5 + 71.8i)T^{2}
89 1+(−6.29+17.3i)T+(−68.1−57.2i)T2 1 + (-6.29 + 17.3i)T + (-68.1 - 57.2i)T^{2}
97 1+(4.43−5.28i)T+(−16.8−95.5i)T2 1 + (4.43 - 5.28i)T + (-16.8 - 95.5i)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

−13.07482231180679303543512066997, −11.89594468628853079833119689944, −10.80078428672623333388142665165, −9.734985002909662170362569711242, −8.623048456520989462616240695639, −7.77569394510154983700580920265, −6.38399005978254315131365728324, −5.90973811349493966543244836810, −3.92052484443969237383655736151, −1.21541994548878993584620893262, 1.90200545587019748011711658207, 3.83484545583305213281099402193, 4.89751354304518334996476133390, 6.98003123486786605352473131954, 7.85843426211155480476401217387, 9.487318723284728690832799457103, 9.788292758988419294752434454636, 10.88888174547306215194455936554, 12.00298418444139303176535939512, 12.60813916181447631343762511840

Graph of the ZZ-function along the critical line