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Properties

Label 2-176-44.43-c5-0-22
Degree 22
Conductor 176176
Sign 0.959+0.280i0.959 + 0.280i
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  + 0.663i·3-s + 68.6·5-s + 214.·7-s + 242.·9-s + (−95.0 − 389. i)11-s − 556. i·13-s + 45.5i·15-s + 1.84e3i·17-s − 1.21e3·19-s + 142. i·21-s − 121. i·23-s + 1.59e3·25-s + 322. i·27-s − 4.26e3i·29-s − 3.81e3i·31-s + ⋯
L(s)  = 1  + 0.0425i·3-s + 1.22·5-s + 1.65·7-s + 0.998·9-s + (−0.236 − 0.971i)11-s − 0.912i·13-s + 0.0523i·15-s + 1.55i·17-s − 0.772·19-s + 0.0704i·21-s − 0.0478i·23-s + 0.509·25-s + 0.0850i·27-s − 0.942i·29-s − 0.712i·31-s + ⋯

Functional equation

Λ(s)=(176s/2 ΓC(s) L(s)=((0.959+0.280i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(176s/2 ΓC(s+5/2) L(s)=((0.959+0.280i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 176176    =    24⋅112^{4} \cdot 11
Sign: 0.959+0.280i0.959 + 0.280i
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), 0.959+0.280i)(2,\ 176,\ (\ :5/2),\ 0.959 + 0.280i)

Particular Values

L(3)L(3) ≈\approx 3.1466044683.146604468
L(12)L(\frac12) ≈\approx 3.1466044683.146604468
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+(95.0+389.i)T 1 + (95.0 + 389. i)T
good3 1−0.663iT−243T2 1 - 0.663iT - 243T^{2}
5 1−68.6T+3.12e3T2 1 - 68.6T + 3.12e3T^{2}
7 1−214.T+1.68e4T2 1 - 214.T + 1.68e4T^{2}
13 1+556.iT−3.71e5T2 1 + 556. iT - 3.71e5T^{2}
17 1−1.84e3iT−1.41e6T2 1 - 1.84e3iT - 1.41e6T^{2}
19 1+1.21e3T+2.47e6T2 1 + 1.21e3T + 2.47e6T^{2}
23 1+121.iT−6.43e6T2 1 + 121. iT - 6.43e6T^{2}
29 1+4.26e3iT−2.05e7T2 1 + 4.26e3iT - 2.05e7T^{2}
31 1+3.81e3iT−2.86e7T2 1 + 3.81e3iT - 2.86e7T^{2}
37 1−9.38e3T+6.93e7T2 1 - 9.38e3T + 6.93e7T^{2}
41 1−1.48e4iT−1.15e8T2 1 - 1.48e4iT - 1.15e8T^{2}
43 1+1.82e4T+1.47e8T2 1 + 1.82e4T + 1.47e8T^{2}
47 1+2.42e4iT−2.29e8T2 1 + 2.42e4iT - 2.29e8T^{2}
53 1−1.80e4T+4.18e8T2 1 - 1.80e4T + 4.18e8T^{2}
59 1−970.iT−7.14e8T2 1 - 970. iT - 7.14e8T^{2}
61 1−4.83e4iT−8.44e8T2 1 - 4.83e4iT - 8.44e8T^{2}
67 1+9.34e3iT−1.35e9T2 1 + 9.34e3iT - 1.35e9T^{2}
71 1−4.24e4iT−1.80e9T2 1 - 4.24e4iT - 1.80e9T^{2}
73 1+2.18e3iT−2.07e9T2 1 + 2.18e3iT - 2.07e9T^{2}
79 1+4.97e4T+3.07e9T2 1 + 4.97e4T + 3.07e9T^{2}
83 1−4.08e4T+3.93e9T2 1 - 4.08e4T + 3.93e9T^{2}
89 1+4.75e3T+5.58e9T2 1 + 4.75e3T + 5.58e9T^{2}
97 1−4.44e4T+8.58e9T2 1 - 4.44e4T + 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

−11.62607935489166799094571656512, −10.59512636448497832502263847967, −10.02543727399954538091470715772, −8.532139669415176399669783672694, −7.88476031032302004235905838197, −6.25093504625275879612135336707, −5.38489657388937217309436291344, −4.14759072734806914158611171812, −2.20582863164656538486421916617, −1.20128831867993346085127433242, 1.47102498887444616750185326078, 2.17829154922091279569060625761, 4.51432447796725327916328229154, 5.13912228732982786792003259594, 6.71918335104891931419054879438, 7.59604171603703176970584659974, 8.981763396671098018425063798523, 9.831111191739539041898371420527, 10.79029142383358395732323646309, 11.84417028925780767111457830514

Graph of the ZZ-function along the critical line