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Properties

Label 2-176-44.43-c5-0-29
Degree 22
Conductor 176176
Sign −0.834−0.551i-0.834 - 0.551i
Analytic cond. 28.227528.2275
Root an. cond. 5.312965.31296
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 26.5i·3-s − 36.9·5-s + 239.·7-s − 463.·9-s + (−334. − 221. i)11-s − 809. i·13-s + 981. i·15-s − 32.3i·17-s − 1.38e3·19-s − 6.37e3i·21-s − 2.85e3i·23-s − 1.76e3·25-s + 5.86e3i·27-s + 5.50e3i·29-s + 3.47e3i·31-s + ⋯
L(s)  = 1  − 1.70i·3-s − 0.660·5-s + 1.85·7-s − 1.90·9-s + (−0.834 − 0.551i)11-s − 1.32i·13-s + 1.12i·15-s − 0.0271i·17-s − 0.882·19-s − 3.15i·21-s − 1.12i·23-s − 0.563·25-s + 1.54i·27-s + 1.21i·29-s + 0.650i·31-s + ⋯

Functional equation

Λ(s)=(176s/2 ΓC(s) L(s)=((−0.834−0.551i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(176s/2 ΓC(s+5/2) L(s)=((−0.834−0.551i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 176176    =    24⋅112^{4} \cdot 11
Sign: −0.834−0.551i-0.834 - 0.551i
Analytic conductor: 28.227528.2275
Root analytic conductor: 5.312965.31296
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ176(175,⋅)\chi_{176} (175, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 176, ( :5/2), −0.834−0.551i)(2,\ 176,\ (\ :5/2),\ -0.834 - 0.551i)

Particular Values

L(3)L(3) ≈\approx 1.1017250691.101725069
L(12)L(\frac12) ≈\approx 1.1017250691.101725069
L(72)L(\frac{7}{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
11 1+(334.+221.i)T 1 + (334. + 221. i)T
good3 1+26.5iT−243T2 1 + 26.5iT - 243T^{2}
5 1+36.9T+3.12e3T2 1 + 36.9T + 3.12e3T^{2}
7 1−239.T+1.68e4T2 1 - 239.T + 1.68e4T^{2}
13 1+809.iT−3.71e5T2 1 + 809. iT - 3.71e5T^{2}
17 1+32.3iT−1.41e6T2 1 + 32.3iT - 1.41e6T^{2}
19 1+1.38e3T+2.47e6T2 1 + 1.38e3T + 2.47e6T^{2}
23 1+2.85e3iT−6.43e6T2 1 + 2.85e3iT - 6.43e6T^{2}
29 1−5.50e3iT−2.05e7T2 1 - 5.50e3iT - 2.05e7T^{2}
31 1−3.47e3iT−2.86e7T2 1 - 3.47e3iT - 2.86e7T^{2}
37 1−4.53e3T+6.93e7T2 1 - 4.53e3T + 6.93e7T^{2}
41 1−5.08e3iT−1.15e8T2 1 - 5.08e3iT - 1.15e8T^{2}
43 1−4.92e3T+1.47e8T2 1 - 4.92e3T + 1.47e8T^{2}
47 1−2.29e4iT−2.29e8T2 1 - 2.29e4iT - 2.29e8T^{2}
53 1+2.91e4T+4.18e8T2 1 + 2.91e4T + 4.18e8T^{2}
59 1+2.19e4iT−7.14e8T2 1 + 2.19e4iT - 7.14e8T^{2}
61 1+4.58e4iT−8.44e8T2 1 + 4.58e4iT - 8.44e8T^{2}
67 1−3.44e4iT−1.35e9T2 1 - 3.44e4iT - 1.35e9T^{2}
71 1−4.48e4iT−1.80e9T2 1 - 4.48e4iT - 1.80e9T^{2}
73 1+7.78e4iT−2.07e9T2 1 + 7.78e4iT - 2.07e9T^{2}
79 1+3.65e4T+3.07e9T2 1 + 3.65e4T + 3.07e9T^{2}
83 1+8.73e4T+3.93e9T2 1 + 8.73e4T + 3.93e9T^{2}
89 1−3.23e4T+5.58e9T2 1 - 3.23e4T + 5.58e9T^{2}
97 1+5.64e3T+8.58e9T2 1 + 5.64e3T + 8.58e9T^{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

−11.25092766470513776955844243682, −10.77667363786424637878241742354, −8.332299085563701470500004290952, −8.173401587359646823939500531801, −7.34471824310460511418831974523, −5.95787616937708341107192528055, −4.81036657012217212531437740192, −2.79669057717730306235889548124, −1.51830898195249978888536874360, −0.34770020171799485587626065041, 2.11742681877453322650657300623, 4.07965047917725690850330917900, 4.49445134775857577540973376391, 5.55119998612726417047264064492, 7.57465773464946439799922231242, 8.424609167784472321516794241052, 9.461917974429044934909071378724, 10.50138967514065633632960139976, 11.36453984169423526257358379034, 11.78840516819466987829211805468

Graph of the ZZ-function along the critical line