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Properties

Label 2-156-12.11-c1-0-21
Degree 22
Conductor 156156
Sign 0.467+0.884i0.467 + 0.884i
Analytic cond. 1.245661.24566
Root an. cond. 1.116091.11609
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  + (0.960 − 1.03i)2-s + (1.58 − 0.687i)3-s + (−0.154 − 1.99i)4-s + 0.716i·5-s + (0.812 − 2.31i)6-s + 4.43i·7-s + (−2.21 − 1.75i)8-s + (2.05 − 2.18i)9-s + (0.743 + 0.687i)10-s − 6.03·11-s + (−1.61 − 3.06i)12-s − 13-s + (4.60 + 4.26i)14-s + (0.492 + 1.13i)15-s + (−3.95 + 0.618i)16-s − 2.94i·17-s + ⋯
L(s)  = 1  + (0.679 − 0.733i)2-s + (0.917 − 0.397i)3-s + (−0.0774 − 0.996i)4-s + 0.320i·5-s + (0.331 − 0.943i)6-s + 1.67i·7-s + (−0.784 − 0.620i)8-s + (0.684 − 0.729i)9-s + (0.235 + 0.217i)10-s − 1.81·11-s + (−0.467 − 0.884i)12-s − 0.277·13-s + (1.23 + 1.13i)14-s + (0.127 + 0.293i)15-s + (−0.987 + 0.154i)16-s − 0.713i·17-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=((0.467+0.884i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s+1/2) L(s)=((0.467+0.884i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: 0.467+0.884i0.467 + 0.884i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ156(131,⋅)\chi_{156} (131, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 156, ( :1/2), 0.467+0.884i)(2,\ 156,\ (\ :1/2),\ 0.467 + 0.884i)

Particular Values

L(1)L(1) ≈\approx 1.57936−0.951855i1.57936 - 0.951855i
L(12)L(\frac12) ≈\approx 1.57936−0.951855i1.57936 - 0.951855i
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+(−0.960+1.03i)T 1 + (-0.960 + 1.03i)T
3 1+(−1.58+0.687i)T 1 + (-1.58 + 0.687i)T
13 1+T 1 + T
good5 1−0.716iT−5T2 1 - 0.716iT - 5T^{2}
7 1−4.43iT−7T2 1 - 4.43iT - 7T^{2}
11 1+6.03T+11T2 1 + 6.03T + 11T^{2}
17 1+2.94iT−17T2 1 + 2.94iT - 17T^{2}
19 1−2.16iT−19T2 1 - 2.16iT - 19T^{2}
23 1−2.19T+23T2 1 - 2.19T + 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+3.96iT−31T2 1 + 3.96iT - 31T^{2}
37 1−4.10T+37T2 1 - 4.10T + 37T^{2}
41 1+6.07iT−41T2 1 + 6.07iT - 41T^{2}
43 1−7.50iT−43T2 1 - 7.50iT - 43T^{2}
47 1−5.69T+47T2 1 - 5.69T + 47T^{2}
53 1−5.88iT−53T2 1 - 5.88iT - 53T^{2}
59 1+1.64T+59T2 1 + 1.64T + 59T^{2}
61 1+4.61T+61T2 1 + 4.61T + 61T^{2}
67 1+11.9iT−67T2 1 + 11.9iT - 67T^{2}
71 1−0.662T+71T2 1 - 0.662T + 71T^{2}
73 1+4.97T+73T2 1 + 4.97T + 73T^{2}
79 1+7.02iT−79T2 1 + 7.02iT - 79T^{2}
83 1+6.68T+83T2 1 + 6.68T + 83T^{2}
89 1−8.94iT−89T2 1 - 8.94iT - 89T^{2}
97 1−8.97T+97T2 1 - 8.97T + 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

−12.73630365763841964349919495866, −12.12018516498643819460928182417, −10.87448893152527634845424028875, −9.720794381400717298732480480385, −8.826518512059402169022260013014, −7.59031517640076498515151293366, −6.03701519471094596629021970624, −4.93619221721226111405565696799, −2.96073653776463315200662963692, −2.37890225537026527212097491878, 2.94059708618874096996455615974, 4.24485580028050687151235782865, 5.12894265262279994750050021954, 7.03325382580897807570531391212, 7.75432627949254005961053312274, 8.651117713411137678286653124093, 10.12645860781462023534052498257, 10.90382172952260660914357922515, 12.84060327274708170735024336743, 13.21116036768967273904336797255

Graph of the ZZ-function along the critical line