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Properties

Label 2-156-13.4-c1-0-2
Degree 22
Conductor 156156
Sign 0.822+0.569i0.822 + 0.569i
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  + (−0.5 + 0.866i)3-s − 4.14i·5-s + (2.08 − 1.20i)7-s + (−0.499 − 0.866i)9-s + (3 + 1.73i)11-s + (1.5 + 3.27i)13-s + (3.58 + 2.07i)15-s + (−2.58 − 4.48i)17-s + (−3 + 1.73i)19-s + 2.41i·21-s + (1 − 1.73i)23-s − 12.1·25-s + 0.999·27-s + (−1.58 + 2.75i)29-s − 1.05i·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s − 1.85i·5-s + (0.789 − 0.455i)7-s + (−0.166 − 0.288i)9-s + (0.904 + 0.522i)11-s + (0.416 + 0.909i)13-s + (0.926 + 0.535i)15-s + (−0.628 − 1.08i)17-s + (−0.688 + 0.397i)19-s + 0.526i·21-s + (0.208 − 0.361i)23-s − 2.43·25-s + 0.192·27-s + (−0.295 + 0.511i)29-s − 0.188i·31-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=((0.822+0.569i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s+1/2) L(s)=((0.822+0.569i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: 0.822+0.569i0.822 + 0.569i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ156(121,⋅)\chi_{156} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 156, ( :1/2), 0.822+0.569i)(2,\ 156,\ (\ :1/2),\ 0.822 + 0.569i)

Particular Values

L(1)L(1) ≈\approx 1.05690−0.330012i1.05690 - 0.330012i
L(12)L(\frac12) ≈\approx 1.05690−0.330012i1.05690 - 0.330012i
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 1
3 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
13 1+(−1.5−3.27i)T 1 + (-1.5 - 3.27i)T
good5 1+4.14iT−5T2 1 + 4.14iT - 5T^{2}
7 1+(−2.08+1.20i)T+(3.5−6.06i)T2 1 + (-2.08 + 1.20i)T + (3.5 - 6.06i)T^{2}
11 1+(−3−1.73i)T+(5.5+9.52i)T2 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2}
17 1+(2.58+4.48i)T+(−8.5+14.7i)T2 1 + (2.58 + 4.48i)T + (-8.5 + 14.7i)T^{2}
19 1+(3−1.73i)T+(9.5−16.4i)T2 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2}
23 1+(−1+1.73i)T+(−11.5−19.9i)T2 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.58−2.75i)T+(−14.5−25.1i)T2 1 + (1.58 - 2.75i)T + (-14.5 - 25.1i)T^{2}
31 1+1.05iT−31T2 1 + 1.05iT - 31T^{2}
37 1+(−6.58−3.80i)T+(18.5+32.0i)T2 1 + (-6.58 - 3.80i)T + (18.5 + 32.0i)T^{2}
41 1+(0.589+0.340i)T+(20.5+35.5i)T2 1 + (0.589 + 0.340i)T + (20.5 + 35.5i)T^{2}
43 1+(−6.08−10.5i)T+(−21.5+37.2i)T2 1 + (-6.08 - 10.5i)T + (-21.5 + 37.2i)T^{2}
47 1−10.3iT−47T2 1 - 10.3iT - 47T^{2}
53 1−1.17T+53T2 1 - 1.17T + 53T^{2}
59 1+(10.1−5.87i)T+(29.5−51.0i)T2 1 + (10.1 - 5.87i)T + (29.5 - 51.0i)T^{2}
61 1+(2.5+4.33i)T+(−30.5+52.8i)T2 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.08+1.20i)T+(33.5+58.0i)T2 1 + (2.08 + 1.20i)T + (33.5 + 58.0i)T^{2}
71 1+(3−1.73i)T+(35.5−61.4i)T2 1 + (3 - 1.73i)T + (35.5 - 61.4i)T^{2}
73 1+14.8iT−73T2 1 + 14.8iT - 73T^{2}
79 1−1.82T+79T2 1 - 1.82T + 79T^{2}
83 1−1.36iT−83T2 1 - 1.36iT - 83T^{2}
89 1+(−6−3.46i)T+(44.5+77.0i)T2 1 + (-6 - 3.46i)T + (44.5 + 77.0i)T^{2}
97 1+(−15.2+8.81i)T+(48.5−84.0i)T2 1 + (-15.2 + 8.81i)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

−12.72404937857673455885675239814, −11.83246236524359221173862133024, −11.03862328200314933896035232560, −9.432009154217241701066443891018, −9.013706066831028281843662931595, −7.80214958935753081860106907376, −6.21014048053753204435227001131, −4.56106484069067626908446634377, −4.47374449864779133404461512491, −1.38926026282112266105857102187, 2.23080141833620993159800757180, 3.73982971399052110874871265241, 5.80147878610528880964109295369, 6.56059910586342399099871911475, 7.66014644328454915840600066703, 8.756441277280242303651446807199, 10.43548644321582412687364247619, 11.05940989265694986908398089580, 11.75164542981706926327522416233, 13.11306510884583046859183370332

Graph of the ZZ-function along the critical line