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Properties

Label 2-176-11.9-c1-0-3
Degree 22
Conductor 176176
Sign −0.273+0.961i-0.273 + 0.961i
Analytic cond. 1.405361.40536
Root an. cond. 1.185481.18548
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  + (−0.743 + 0.540i)3-s + (−1.24 − 3.82i)5-s + (−3.01 − 2.18i)7-s + (−0.665 + 2.04i)9-s + (1.97 − 2.66i)11-s + (0.324 − 0.998i)13-s + (2.99 + 2.17i)15-s + (−0.291 − 0.898i)17-s + (1.92 − 1.40i)19-s + 3.42·21-s + 1.73·23-s + (−9.05 + 6.58i)25-s + (−1.46 − 4.50i)27-s + (2.22 + 1.61i)29-s + (−1.47 + 4.55i)31-s + ⋯
L(s)  = 1  + (−0.429 + 0.311i)3-s + (−0.556 − 1.71i)5-s + (−1.13 − 0.827i)7-s + (−0.221 + 0.683i)9-s + (0.594 − 0.804i)11-s + (0.0899 − 0.276i)13-s + (0.772 + 0.561i)15-s + (−0.0708 − 0.217i)17-s + (0.442 − 0.321i)19-s + 0.746·21-s + 0.362·23-s + (−1.81 + 1.31i)25-s + (−0.281 − 0.867i)27-s + (0.412 + 0.300i)29-s + (−0.265 + 0.817i)31-s + ⋯

Functional equation

Λ(s)=(176s/2 ΓC(s) L(s)=((−0.273+0.961i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(176s/2 ΓC(s+1/2) L(s)=((−0.273+0.961i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 176176    =    24⋅112^{4} \cdot 11
Sign: −0.273+0.961i-0.273 + 0.961i
Analytic conductor: 1.405361.40536
Root analytic conductor: 1.185481.18548
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ176(97,⋅)\chi_{176} (97, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 176, ( :1/2), −0.273+0.961i)(2,\ 176,\ (\ :1/2),\ -0.273 + 0.961i)

Particular Values

L(1)L(1) ≈\approx 0.417239−0.552243i0.417239 - 0.552243i
L(12)L(\frac12) ≈\approx 0.417239−0.552243i0.417239 - 0.552243i
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
11 1+(−1.97+2.66i)T 1 + (-1.97 + 2.66i)T
good3 1+(0.743−0.540i)T+(0.927−2.85i)T2 1 + (0.743 - 0.540i)T + (0.927 - 2.85i)T^{2}
5 1+(1.24+3.82i)T+(−4.04+2.93i)T2 1 + (1.24 + 3.82i)T + (-4.04 + 2.93i)T^{2}
7 1+(3.01+2.18i)T+(2.16+6.65i)T2 1 + (3.01 + 2.18i)T + (2.16 + 6.65i)T^{2}
13 1+(−0.324+0.998i)T+(−10.5−7.64i)T2 1 + (-0.324 + 0.998i)T + (-10.5 - 7.64i)T^{2}
17 1+(0.291+0.898i)T+(−13.7+9.99i)T2 1 + (0.291 + 0.898i)T + (-13.7 + 9.99i)T^{2}
19 1+(−1.92+1.40i)T+(5.87−18.0i)T2 1 + (-1.92 + 1.40i)T + (5.87 - 18.0i)T^{2}
23 1−1.73T+23T2 1 - 1.73T + 23T^{2}
29 1+(−2.22−1.61i)T+(8.96+27.5i)T2 1 + (-2.22 - 1.61i)T + (8.96 + 27.5i)T^{2}
31 1+(1.47−4.55i)T+(−25.0−18.2i)T2 1 + (1.47 - 4.55i)T + (-25.0 - 18.2i)T^{2}
37 1+(1.71+1.24i)T+(11.4+35.1i)T2 1 + (1.71 + 1.24i)T + (11.4 + 35.1i)T^{2}
41 1+(−7.38+5.36i)T+(12.6−38.9i)T2 1 + (-7.38 + 5.36i)T + (12.6 - 38.9i)T^{2}
43 1+0.431T+43T2 1 + 0.431T + 43T^{2}
47 1+(−5.11+3.71i)T+(14.5−44.6i)T2 1 + (-5.11 + 3.71i)T + (14.5 - 44.6i)T^{2}
53 1+(−0.0976+0.300i)T+(−42.8−31.1i)T2 1 + (-0.0976 + 0.300i)T + (-42.8 - 31.1i)T^{2}
59 1+(−6.54−4.75i)T+(18.2+56.1i)T2 1 + (-6.54 - 4.75i)T + (18.2 + 56.1i)T^{2}
61 1+(4.21+12.9i)T+(−49.3+35.8i)T2 1 + (4.21 + 12.9i)T + (-49.3 + 35.8i)T^{2}
67 1+5.68T+67T2 1 + 5.68T + 67T^{2}
71 1+(−3.97−12.2i)T+(−57.4+41.7i)T2 1 + (-3.97 - 12.2i)T + (-57.4 + 41.7i)T^{2}
73 1+(−2.84−2.06i)T+(22.5+69.4i)T2 1 + (-2.84 - 2.06i)T + (22.5 + 69.4i)T^{2}
79 1+(−2.73+8.40i)T+(−63.9−46.4i)T2 1 + (-2.73 + 8.40i)T + (-63.9 - 46.4i)T^{2}
83 1+(4.53+13.9i)T+(−67.1+48.7i)T2 1 + (4.53 + 13.9i)T + (-67.1 + 48.7i)T^{2}
89 1−2.43T+89T2 1 - 2.43T + 89T^{2}
97 1+(−0.457+1.40i)T+(−78.4−57.0i)T2 1 + (-0.457 + 1.40i)T + (-78.4 - 57.0i)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

−12.46841480913431735065278230464, −11.48514392381372088551512748830, −10.46907654960237223920439689651, −9.293048451175214255501297095316, −8.502499954918004623434601620660, −7.26184435705957710173773610563, −5.77028632117067388605651632994, −4.73325105820893329435214122761, −3.59167819931955364573235596547, −0.67812747072290170690573639269, 2.71970576039742230833987591170, 3.81931505250471424474722666511, 6.04229052329501929335123120022, 6.58770770870731449405398288657, 7.49317031383483779192281630653, 9.176055859669055409328542619834, 10.01997635704176197463276690571, 11.23265190997190256127775143216, 11.94117014176327566446386236522, 12.69815887555748676302491784615

Graph of the ZZ-function along the critical line