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Properties

Label 2-176-11.6-c2-0-5
Degree 22
Conductor 176176
Sign 0.805−0.592i0.805 - 0.592i
Analytic cond. 4.795654.79565
Root an. cond. 2.189892.18989
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.08 + 2.97i)3-s + (1.46 − 4.50i)5-s + (3.91 + 5.39i)7-s + (5.10 + 15.7i)9-s + (−6.32 − 9.00i)11-s + (14.3 − 4.65i)13-s + (19.3 − 14.0i)15-s + (−16.9 − 5.52i)17-s + (−19.0 + 26.1i)19-s + 33.6i·21-s + 3.58·23-s + (2.04 + 1.48i)25-s + (−11.7 + 36.2i)27-s + (−7.50 − 10.3i)29-s + (−6.76 − 20.8i)31-s + ⋯
L(s)  = 1  + (1.36 + 0.990i)3-s + (0.293 − 0.901i)5-s + (0.559 + 0.770i)7-s + (0.567 + 1.74i)9-s + (−0.574 − 0.818i)11-s + (1.10 − 0.357i)13-s + (1.29 − 0.938i)15-s + (−0.999 − 0.324i)17-s + (−1.00 + 1.37i)19-s + 1.60i·21-s + 0.156·23-s + (0.0816 + 0.0592i)25-s + (−0.435 + 1.34i)27-s + (−0.258 − 0.356i)29-s + (−0.218 − 0.671i)31-s + ⋯

Functional equation

Λ(s)=(176s/2 ΓC(s) L(s)=((0.805−0.592i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(176s/2 ΓC(s+1) L(s)=((0.805−0.592i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 176176    =    24⋅112^{4} \cdot 11
Sign: 0.805−0.592i0.805 - 0.592i
Analytic conductor: 4.795654.79565
Root analytic conductor: 2.189892.18989
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ176(17,⋅)\chi_{176} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 176, ( :1), 0.805−0.592i)(2,\ 176,\ (\ :1),\ 0.805 - 0.592i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.27100+0.745233i2.27100 + 0.745233i
L(12)L(\frac12) ≈\approx 2.27100+0.745233i2.27100 + 0.745233i
L(2)L(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+(6.32+9.00i)T 1 + (6.32 + 9.00i)T
good3 1+(−4.08−2.97i)T+(2.78+8.55i)T2 1 + (-4.08 - 2.97i)T + (2.78 + 8.55i)T^{2}
5 1+(−1.46+4.50i)T+(−20.2−14.6i)T2 1 + (-1.46 + 4.50i)T + (-20.2 - 14.6i)T^{2}
7 1+(−3.91−5.39i)T+(−15.1+46.6i)T2 1 + (-3.91 - 5.39i)T + (-15.1 + 46.6i)T^{2}
13 1+(−14.3+4.65i)T+(136.−99.3i)T2 1 + (-14.3 + 4.65i)T + (136. - 99.3i)T^{2}
17 1+(16.9+5.52i)T+(233.+169.i)T2 1 + (16.9 + 5.52i)T + (233. + 169. i)T^{2}
19 1+(19.0−26.1i)T+(−111.−343.i)T2 1 + (19.0 - 26.1i)T + (-111. - 343. i)T^{2}
23 1−3.58T+529T2 1 - 3.58T + 529T^{2}
29 1+(7.50+10.3i)T+(−259.+799.i)T2 1 + (7.50 + 10.3i)T + (-259. + 799. i)T^{2}
31 1+(6.76+20.8i)T+(−777.+564.i)T2 1 + (6.76 + 20.8i)T + (-777. + 564. i)T^{2}
37 1+(−22.4+16.2i)T+(423.−1.30e3i)T2 1 + (-22.4 + 16.2i)T + (423. - 1.30e3i)T^{2}
41 1+(29.1−40.0i)T+(−519.−1.59e3i)T2 1 + (29.1 - 40.0i)T + (-519. - 1.59e3i)T^{2}
43 1+16.8iT−1.84e3T2 1 + 16.8iT - 1.84e3T^{2}
47 1+(63.0+45.8i)T+(682.+2.10e3i)T2 1 + (63.0 + 45.8i)T + (682. + 2.10e3i)T^{2}
53 1+(−3.06−9.41i)T+(−2.27e3+1.65e3i)T2 1 + (-3.06 - 9.41i)T + (-2.27e3 + 1.65e3i)T^{2}
59 1+(−29.6+21.5i)T+(1.07e3−3.31e3i)T2 1 + (-29.6 + 21.5i)T + (1.07e3 - 3.31e3i)T^{2}
61 1+(26.9+8.74i)T+(3.01e3+2.18e3i)T2 1 + (26.9 + 8.74i)T + (3.01e3 + 2.18e3i)T^{2}
67 1−30.7T+4.48e3T2 1 - 30.7T + 4.48e3T^{2}
71 1+(−12.6+39.0i)T+(−4.07e3−2.96e3i)T2 1 + (-12.6 + 39.0i)T + (-4.07e3 - 2.96e3i)T^{2}
73 1+(36.3+49.9i)T+(−1.64e3+5.06e3i)T2 1 + (36.3 + 49.9i)T + (-1.64e3 + 5.06e3i)T^{2}
79 1+(118.−38.4i)T+(5.04e3−3.66e3i)T2 1 + (118. - 38.4i)T + (5.04e3 - 3.66e3i)T^{2}
83 1+(−2.94−0.956i)T+(5.57e3+4.04e3i)T2 1 + (-2.94 - 0.956i)T + (5.57e3 + 4.04e3i)T^{2}
89 1−60.4T+7.92e3T2 1 - 60.4T + 7.92e3T^{2}
97 1+(−14.8−45.6i)T+(−7.61e3+5.53e3i)T2 1 + (-14.8 - 45.6i)T + (-7.61e3 + 5.53e3i)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

−12.98291743569626937401008683198, −11.40418653338596517910353302985, −10.40026893286873565120121287004, −9.278940532613401237360451695288, −8.475948375288388748898390020549, −8.189149315397245042566738156639, −5.88538369297446988395421980117, −4.76169769578713731810963550185, −3.54615376212845005359259548791, −2.07956297410199193584629502119, 1.74367606147749083446988587576, 2.86643998241295459988294250665, 4.35495673973311382728311799356, 6.61991885246444771806974354288, 7.10040815286337837435642394803, 8.230332821372581660278299945315, 9.052329134665939077841640872126, 10.46132106211160510203878330661, 11.23564408022543122798404659632, 12.90987031925461686090346181936

Graph of the ZZ-function along the critical line