Properties

Label 2-152-152.51-c1-0-15
Degree 22
Conductor 152152
Sign 0.973+0.230i0.973 + 0.230i
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 + (1.18 − 3.24i)3-s + (−0.347 + 1.96i)4-s + (4.58 − 1.66i)6-s + (−2.44 + 1.41i)8-s + (−6.83 − 5.73i)9-s + (2.33 + 4.04i)11-s + (5.97 + 3.45i)12-s + (−3.75 − 1.36i)16-s + (−1.45 + 1.22i)17-s − 12.6i·18-s + (4.00 + 1.72i)19-s + (−2.25 + 6.20i)22-s + (1.69 + 9.61i)24-s + (−4.69 + 1.71i)25-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (0.681 − 1.87i)3-s + (−0.173 + 0.984i)4-s + (1.87 − 0.681i)6-s + (−0.866 + 0.500i)8-s + (−2.27 − 1.91i)9-s + (0.704 + 1.21i)11-s + (1.72 + 0.996i)12-s + (−0.939 − 0.342i)16-s + (−0.352 + 0.296i)17-s − 2.97i·18-s + (0.918 + 0.394i)19-s + (−0.481 + 1.32i)22-s + (0.346 + 1.96i)24-s + (−0.939 + 0.342i)25-s + ⋯

Functional equation

Λ(s)=(152s/2ΓC(s)L(s)=((0.973+0.230i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(152s/2ΓC(s+1/2)L(s)=((0.973+0.230i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 152152    =    23192^{3} \cdot 19
Sign: 0.973+0.230i0.973 + 0.230i
Analytic conductor: 1.213721.21372
Root analytic conductor: 1.101691.10169
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ152(51,)\chi_{152} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 152, ( :1/2), 0.973+0.230i)(2,\ 152,\ (\ :1/2),\ 0.973 + 0.230i)

Particular Values

L(1)L(1) \approx 1.701530.198380i1.70153 - 0.198380i
L(12)L(\frac12) \approx 1.701530.198380i1.70153 - 0.198380i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.9091.08i)T 1 + (-0.909 - 1.08i)T
19 1+(4.001.72i)T 1 + (-4.00 - 1.72i)T
good3 1+(1.18+3.24i)T+(2.291.92i)T2 1 + (-1.18 + 3.24i)T + (-2.29 - 1.92i)T^{2}
5 1+(4.691.71i)T2 1 + (4.69 - 1.71i)T^{2}
7 1+(3.5+6.06i)T2 1 + (3.5 + 6.06i)T^{2}
11 1+(2.334.04i)T+(5.5+9.52i)T2 1 + (-2.33 - 4.04i)T + (-5.5 + 9.52i)T^{2}
13 1+(9.958.35i)T2 1 + (9.95 - 8.35i)T^{2}
17 1+(1.451.22i)T+(2.9516.7i)T2 1 + (1.45 - 1.22i)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+(15.526.8i)T2 1 + (-15.5 - 26.8i)T^{2}
37 1+37T2 1 + 37T^{2}
41 1+(1.34+3.70i)T+(31.426.3i)T2 1 + (-1.34 + 3.70i)T + (-31.4 - 26.3i)T^{2}
43 1+(2.14+12.1i)T+(40.4+14.7i)T2 1 + (2.14 + 12.1i)T + (-40.4 + 14.7i)T^{2}
47 1+(8.1646.2i)T2 1 + (-8.16 - 46.2i)T^{2}
53 1+(49.818.1i)T2 1 + (-49.8 - 18.1i)T^{2}
59 1+(9.8611.7i)T+(10.2+58.1i)T2 1 + (-9.86 - 11.7i)T + (-10.2 + 58.1i)T^{2}
61 1+(57.3+20.8i)T2 1 + (57.3 + 20.8i)T^{2}
67 1+(7.91+9.43i)T+(11.665.9i)T2 1 + (-7.91 + 9.43i)T + (-11.6 - 65.9i)T^{2}
71 1+(66.7+24.2i)T2 1 + (-66.7 + 24.2i)T^{2}
73 1+(7.22+2.62i)T+(55.9+46.9i)T2 1 + (7.22 + 2.62i)T + (55.9 + 46.9i)T^{2}
79 1+(60.5+50.7i)T2 1 + (60.5 + 50.7i)T^{2}
83 1+(0.170+0.294i)T+(41.571.8i)T2 1 + (-0.170 + 0.294i)T + (-41.5 - 71.8i)T^{2}
89 1+(4.3611.9i)T+(68.1+57.2i)T2 1 + (-4.36 - 11.9i)T + (-68.1 + 57.2i)T^{2}
97 1+(8.22+9.80i)T+(16.8+95.5i)T2 1 + (8.22 + 9.80i)T + (-16.8 + 95.5i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−13.14880359193795375547832457354, −12.20590905229838886927646644991, −11.76390382854928949498166392594, −9.375964422101208504678643954668, −8.365753861823423950510282181136, −7.37457657094967253590996987715, −6.82149037340385372951350959510, −5.66210075499383552096713239828, −3.67489148035696175270220736082, −2.05880234418347244395018623740, 2.86695587920878528688999874646, 3.79912227473024341795502571271, 4.85040987597856448870964717988, 5.96451182560450691026781664244, 8.348823173270527279524169261967, 9.352557615455495874950149151719, 9.932572800734602349222443116711, 11.18382566105165015213552586650, 11.48557251552763405025148352676, 13.35692912592296562357809572104

Graph of the ZZ-function along the critical line