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Properties

Label 2-672-4.3-c2-0-21
Degree 22
Conductor 672672
Sign −0.707+0.707i-0.707 + 0.707i
Analytic cond. 18.310618.3106
Root an. cond. 4.279094.27909
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  + 1.73i·3-s + 1.08·5-s − 2.64i·7-s − 2.99·9-s − 3.75i·11-s − 10.2·13-s + 1.88i·15-s − 19.5·17-s − 9.33i·19-s + 4.58·21-s − 1.06i·23-s − 23.8·25-s − 5.19i·27-s − 0.156·29-s + 15.1i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.217·5-s − 0.377i·7-s − 0.333·9-s − 0.341i·11-s − 0.787·13-s + 0.125i·15-s − 1.15·17-s − 0.491i·19-s + 0.218·21-s − 0.0462i·23-s − 0.952·25-s − 0.192i·27-s − 0.00540·29-s + 0.487i·31-s + ⋯

Functional equation

Λ(s)=(672s/2 ΓC(s) L(s)=((−0.707+0.707i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(672s/2 ΓC(s+1) L(s)=((−0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 672672    =    25⋅3⋅72^{5} \cdot 3 \cdot 7
Sign: −0.707+0.707i-0.707 + 0.707i
Analytic conductor: 18.310618.3106
Root analytic conductor: 4.279094.27909
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ672(127,⋅)\chi_{672} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 672, ( :1), −0.707+0.707i)(2,\ 672,\ (\ :1),\ -0.707 + 0.707i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.38950637790.3895063779
L(12)L(\frac12) ≈\approx 0.38950637790.3895063779
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
3 1−1.73iT 1 - 1.73iT
7 1+2.64iT 1 + 2.64iT
good5 1−1.08T+25T2 1 - 1.08T + 25T^{2}
11 1+3.75iT−121T2 1 + 3.75iT - 121T^{2}
13 1+10.2T+169T2 1 + 10.2T + 169T^{2}
17 1+19.5T+289T2 1 + 19.5T + 289T^{2}
19 1+9.33iT−361T2 1 + 9.33iT - 361T^{2}
23 1+1.06iT−529T2 1 + 1.06iT - 529T^{2}
29 1+0.156T+841T2 1 + 0.156T + 841T^{2}
31 1−15.1iT−961T2 1 - 15.1iT - 961T^{2}
37 1+20.0T+1.36e3T2 1 + 20.0T + 1.36e3T^{2}
41 1+43.2T+1.68e3T2 1 + 43.2T + 1.68e3T^{2}
43 1+21.0iT−1.84e3T2 1 + 21.0iT - 1.84e3T^{2}
47 1+21.8iT−2.20e3T2 1 + 21.8iT - 2.20e3T^{2}
53 1−30.6T+2.80e3T2 1 - 30.6T + 2.80e3T^{2}
59 1+59.5iT−3.48e3T2 1 + 59.5iT - 3.48e3T^{2}
61 1+58.7T+3.72e3T2 1 + 58.7T + 3.72e3T^{2}
67 1+84.8iT−4.48e3T2 1 + 84.8iT - 4.48e3T^{2}
71 1+12.1iT−5.04e3T2 1 + 12.1iT - 5.04e3T^{2}
73 1+48.8T+5.32e3T2 1 + 48.8T + 5.32e3T^{2}
79 1−106.iT−6.24e3T2 1 - 106. iT - 6.24e3T^{2}
83 1+128.iT−6.88e3T2 1 + 128. iT - 6.88e3T^{2}
89 1−100.T+7.92e3T2 1 - 100.T + 7.92e3T^{2}
97 1+101.T+9.40e3T2 1 + 101.T + 9.40e3T^{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

−10.01131790670100665334506842938, −9.190189255623221359570377629380, −8.397992631254137101522585787946, −7.28613672527585826505093666292, −6.41538130825814747978857733884, −5.27176386577211730897443187749, −4.44928951137695548684045828961, −3.34778508429709977583931786418, −2.06818242760956466677166614096, −0.12871883228029944496751002013, 1.73865454077464573299281120277, 2.68731140514758713753568139053, 4.17255738505170895016498164663, 5.29192175731649400237073299840, 6.22960163342259261590181827156, 7.10699695096288397269573672522, 7.954869099985049841923537714136, 8.896699141819923601516681544670, 9.710078996135071020645230246254, 10.59696412099043406696139606946

Graph of the ZZ-function along the critical line