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Properties

Label 2-588-12.11-c1-0-32
Degree 22
Conductor 588588
Sign 0.998+0.0547i0.998 + 0.0547i
Analytic cond. 4.695204.69520
Root an. cond. 2.166842.16684
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  + (0.446 + 1.34i)2-s + (−1.32 − 1.11i)3-s + (−1.60 + 1.19i)4-s − 0.803i·5-s + (0.899 − 2.27i)6-s + (−2.32 − 1.61i)8-s + (0.523 + 2.95i)9-s + (1.07 − 0.358i)10-s − 2.34·11-s + (3.45 + 0.189i)12-s + 5.26·13-s + (−0.893 + 1.06i)15-s + (1.12 − 3.83i)16-s − 1.18i·17-s + (−3.72 + 2.02i)18-s − 7.12i·19-s + ⋯
L(s)  = 1  + (0.315 + 0.948i)2-s + (−0.766 − 0.642i)3-s + (−0.800 + 0.599i)4-s − 0.359i·5-s + (0.367 − 0.930i)6-s + (−0.821 − 0.569i)8-s + (0.174 + 0.984i)9-s + (0.340 − 0.113i)10-s − 0.707·11-s + (0.998 + 0.0547i)12-s + 1.46·13-s + (−0.230 + 0.275i)15-s + (0.281 − 0.959i)16-s − 0.287i·17-s + (−0.879 + 0.476i)18-s − 1.63i·19-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((0.998+0.0547i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0547i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+1/2) L(s)=((0.998+0.0547i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.998+0.0547i0.998 + 0.0547i
Analytic conductor: 4.695204.69520
Root analytic conductor: 2.166842.16684
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ588(491,⋅)\chi_{588} (491, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :1/2), 0.998+0.0547i)(2,\ 588,\ (\ :1/2),\ 0.998 + 0.0547i)

Particular Values

L(1)L(1) ≈\approx 1.15230−0.0315413i1.15230 - 0.0315413i
L(12)L(\frac12) ≈\approx 1.15230−0.0315413i1.15230 - 0.0315413i
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+(−0.446−1.34i)T 1 + (-0.446 - 1.34i)T
3 1+(1.32+1.11i)T 1 + (1.32 + 1.11i)T
7 1 1
good5 1+0.803iT−5T2 1 + 0.803iT - 5T^{2}
11 1+2.34T+11T2 1 + 2.34T + 11T^{2}
13 1−5.26T+13T2 1 - 5.26T + 13T^{2}
17 1+1.18iT−17T2 1 + 1.18iT - 17T^{2}
19 1+7.12iT−19T2 1 + 7.12iT - 19T^{2}
23 1−7.88T+23T2 1 - 7.88T + 23T^{2}
29 1+4.23iT−29T2 1 + 4.23iT - 29T^{2}
31 1+4.89iT−31T2 1 + 4.89iT - 31T^{2}
37 1−1.04T+37T2 1 - 1.04T + 37T^{2}
41 1−7.16iT−41T2 1 - 7.16iT - 41T^{2}
43 1−7.94iT−43T2 1 - 7.94iT - 43T^{2}
47 1+6.09T+47T2 1 + 6.09T + 47T^{2}
53 1+8.72iT−53T2 1 + 8.72iT - 53T^{2}
59 1+0.662T+59T2 1 + 0.662T + 59T^{2}
61 1−0.958T+61T2 1 - 0.958T + 61T^{2}
67 1+8.42iT−67T2 1 + 8.42iT - 67T^{2}
71 1−9.67T+71T2 1 - 9.67T + 71T^{2}
73 1+1.41T+73T2 1 + 1.41T + 73T^{2}
79 1−6.92iT−79T2 1 - 6.92iT - 79T^{2}
83 1+5.18T+83T2 1 + 5.18T + 83T^{2}
89 1+16.3iT−89T2 1 + 16.3iT - 89T^{2}
97 1+4.37T+97T2 1 + 4.37T + 97T^{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

−11.04397459639656231711413962088, −9.603406983085260125225724394421, −8.629603565724761381773538574368, −7.893593993890682937306174450634, −6.89282329662253449901573288680, −6.26035133915951681315631619437, −5.20299547935448570402140308101, −4.59410065390423805430276284778, −2.94175952503214804468344791840, −0.78347227996802227378881050278, 1.27295861066982493299156417910, 3.13551205661835194280870846020, 3.88403758750064425494718174365, 5.10325507741140256050094446649, 5.78791793991457463413701508141, 6.82550997070366373903524209071, 8.441049394588522196356855305936, 9.156352540532201046480336362544, 10.41693297020776204787083691312, 10.60128364881388564050739062822

Graph of the ZZ-function along the critical line