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Properties

Label 2-968-11.8-c0-0-0
Degree 22
Conductor 968968
Sign 0.0560+0.998i0.0560 + 0.998i
Analytic cond. 0.4830940.483094
Root an. cond. 0.6950500.695050
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)5-s + (1.34 + 0.437i)7-s + (−0.309 + 0.951i)15-s + (0.831 − 1.14i)17-s − 1.41i·21-s − 23-s + (−0.809 − 0.587i)27-s + (−0.809 + 0.587i)31-s + (−0.831 − 1.14i)35-s + (0.309 − 0.951i)37-s + 1.41i·43-s + (0.809 + 0.587i)49-s + (−1.34 − 0.437i)51-s + (0.309 − 0.951i)59-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)5-s + (1.34 + 0.437i)7-s + (−0.309 + 0.951i)15-s + (0.831 − 1.14i)17-s − 1.41i·21-s − 23-s + (−0.809 − 0.587i)27-s + (−0.809 + 0.587i)31-s + (−0.831 − 1.14i)35-s + (0.309 − 0.951i)37-s + 1.41i·43-s + (0.809 + 0.587i)49-s + (−1.34 − 0.437i)51-s + (0.309 − 0.951i)59-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=((0.0560+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0560 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s) L(s)=((0.0560+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0560 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: 0.0560+0.998i0.0560 + 0.998i
Analytic conductor: 0.4830940.483094
Root analytic conductor: 0.6950500.695050
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ968(481,⋅)\chi_{968} (481, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 968, ( :0), 0.0560+0.998i)(2,\ 968,\ (\ :0),\ 0.0560 + 0.998i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.91946723370.9194672337
L(12)L(\frac12) ≈\approx 0.91946723370.9194672337
L(1)L(1) 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
good3 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
5 1+(0.809+0.587i)T+(0.309+0.951i)T2 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2}
7 1+(−1.34−0.437i)T+(0.809+0.587i)T2 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2}
13 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
17 1+(−0.831+1.14i)T+(−0.309−0.951i)T2 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2}
19 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
23 1+T+T2 1 + T + T^{2}
29 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
31 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
37 1+(−0.309+0.951i)T+(−0.809−0.587i)T2 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2}
41 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
43 1−1.41iT−T2 1 - 1.41iT - T^{2}
47 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
53 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
59 1+(−0.309+0.951i)T+(−0.809−0.587i)T2 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2}
61 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
67 1−T+T2 1 - T + T^{2}
71 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
73 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
79 1+(−0.831−1.14i)T+(−0.309+0.951i)T2 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2}
83 1+(0.831−1.14i)T+(−0.309−0.951i)T2 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2}
89 1+T+T2 1 + T + T^{2}
97 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)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

−9.976122158878763470259100163074, −9.009401566965570370341194512573, −7.925101541511993234645651953449, −7.82702010236782684703440734912, −6.76363277597471891071497302005, −5.61302499055982326930347013028, −4.86275841968163416794270232086, −3.84433360353964124388669971458, −2.21110414080452641908912603621, −1.04052006312900831254152696419, 1.79232013696017646061738422107, 3.58418886820362785195632855123, 4.13077820116991630063198439689, 5.02798958301002478672233356594, 5.95616984084919423972195366782, 7.33957599405351662728484509097, 7.82294678613158328617213124504, 8.653850844395604142085851973840, 9.935796356815212136996426776718, 10.44781039295252739501896378935

Graph of the ZZ-function along the critical line