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Properties

Label 2-176-11.3-c1-0-0
Degree 22
Conductor 176176
Sign 0.205−0.978i0.205 - 0.978i
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  + (0.632 + 1.94i)3-s + (0.132 + 0.0962i)5-s + (−1.08 + 3.32i)7-s + (−0.962 + 0.699i)9-s + (0.605 − 3.26i)11-s + (−2.17 + 1.58i)13-s + (−0.103 + 0.318i)15-s + (6.12 + 4.45i)17-s + (−1.42 − 4.39i)19-s − 7.16·21-s + 0.706·23-s + (−1.53 − 4.72i)25-s + (2.99 + 2.17i)27-s + (−0.317 + 0.978i)29-s + (4.36 − 3.17i)31-s + ⋯
L(s)  = 1  + (0.365 + 1.12i)3-s + (0.0592 + 0.0430i)5-s + (−0.408 + 1.25i)7-s + (−0.320 + 0.233i)9-s + (0.182 − 0.983i)11-s + (−0.604 + 0.439i)13-s + (−0.0267 + 0.0823i)15-s + (1.48 + 1.08i)17-s + (−0.327 − 1.00i)19-s − 1.56·21-s + 0.147·23-s + (−0.307 − 0.945i)25-s + (0.577 + 0.419i)27-s + (−0.0590 + 0.181i)29-s + (0.784 − 0.570i)31-s + ⋯

Functional equation

Λ(s)=(176s/2 ΓC(s) L(s)=((0.205−0.978i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(176s/2 ΓC(s+1/2) L(s)=((0.205−0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 176176    =    24⋅112^{4} \cdot 11
Sign: 0.205−0.978i0.205 - 0.978i
Analytic conductor: 1.405361.40536
Root analytic conductor: 1.185481.18548
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ176(113,⋅)\chi_{176} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 176, ( :1/2), 0.205−0.978i)(2,\ 176,\ (\ :1/2),\ 0.205 - 0.978i)

Particular Values

L(1)L(1) ≈\approx 0.978595+0.794186i0.978595 + 0.794186i
L(12)L(\frac12) ≈\approx 0.978595+0.794186i0.978595 + 0.794186i
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+(−0.605+3.26i)T 1 + (-0.605 + 3.26i)T
good3 1+(−0.632−1.94i)T+(−2.42+1.76i)T2 1 + (-0.632 - 1.94i)T + (-2.42 + 1.76i)T^{2}
5 1+(−0.132−0.0962i)T+(1.54+4.75i)T2 1 + (-0.132 - 0.0962i)T + (1.54 + 4.75i)T^{2}
7 1+(1.08−3.32i)T+(−5.66−4.11i)T2 1 + (1.08 - 3.32i)T + (-5.66 - 4.11i)T^{2}
13 1+(2.17−1.58i)T+(4.01−12.3i)T2 1 + (2.17 - 1.58i)T + (4.01 - 12.3i)T^{2}
17 1+(−6.12−4.45i)T+(5.25+16.1i)T2 1 + (-6.12 - 4.45i)T + (5.25 + 16.1i)T^{2}
19 1+(1.42+4.39i)T+(−15.3+11.1i)T2 1 + (1.42 + 4.39i)T + (-15.3 + 11.1i)T^{2}
23 1−0.706T+23T2 1 - 0.706T + 23T^{2}
29 1+(0.317−0.978i)T+(−23.4−17.0i)T2 1 + (0.317 - 0.978i)T + (-23.4 - 17.0i)T^{2}
31 1+(−4.36+3.17i)T+(9.57−29.4i)T2 1 + (-4.36 + 3.17i)T + (9.57 - 29.4i)T^{2}
37 1+(−3.58+11.0i)T+(−29.9−21.7i)T2 1 + (-3.58 + 11.0i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.0867+0.267i)T+(−33.1+24.0i)T2 1 + (0.0867 + 0.267i)T + (-33.1 + 24.0i)T^{2}
43 1+4.31T+43T2 1 + 4.31T + 43T^{2}
47 1+(1.80+5.54i)T+(−38.0+27.6i)T2 1 + (1.80 + 5.54i)T + (-38.0 + 27.6i)T^{2}
53 1+(7.98−5.80i)T+(16.3−50.4i)T2 1 + (7.98 - 5.80i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.954+2.93i)T+(−47.7−34.6i)T2 1 + (-0.954 + 2.93i)T + (-47.7 - 34.6i)T^{2}
61 1+(5.60+4.07i)T+(18.8+58.0i)T2 1 + (5.60 + 4.07i)T + (18.8 + 58.0i)T^{2}
67 1+1.91T+67T2 1 + 1.91T + 67T^{2}
71 1+(−9.84−7.15i)T+(21.9+67.5i)T2 1 + (-9.84 - 7.15i)T + (21.9 + 67.5i)T^{2}
73 1+(3.51−10.8i)T+(−59.0−42.9i)T2 1 + (3.51 - 10.8i)T + (-59.0 - 42.9i)T^{2}
79 1+(1.39−1.01i)T+(24.4−75.1i)T2 1 + (1.39 - 1.01i)T + (24.4 - 75.1i)T^{2}
83 1+(−3.81−2.76i)T+(25.6+78.9i)T2 1 + (-3.81 - 2.76i)T + (25.6 + 78.9i)T^{2}
89 1−6.31T+89T2 1 - 6.31T + 89T^{2}
97 1+(5.66−4.11i)T+(29.9−92.2i)T2 1 + (5.66 - 4.11i)T + (29.9 - 92.2i)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

−12.78910917917367609455976449908, −11.93783356083992030203952430430, −10.78545077875222608301362614272, −9.755851615318848197798348598797, −9.079029026967876296918724657305, −8.152225243586906976660095379235, −6.37491048399335312868704848626, −5.33896613363675125677552180647, −3.93642021275947198122313611779, −2.71849584237390381850445638510, 1.35581194759338434609970806089, 3.16636848561437883201834090916, 4.80020257559731952294416070601, 6.49830332980487890222439246367, 7.44702989004712179308848779685, 7.903697349301780603942310972774, 9.703539046774580095873210925889, 10.24371291211192062639792896442, 11.85734630982709314041070103355, 12.58892775332025253809914477572

Graph of the ZZ-function along the critical line