Properties

Label 2-1640-41.40-c1-0-1
Degree 22
Conductor 16401640
Sign 0.6720.740i0.672 - 0.740i
Analytic cond. 13.095413.0954
Root an. cond. 3.618763.61876
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  − 2.65i·3-s − 5-s − 1.97i·7-s − 4.04·9-s + 5.81i·11-s + 1.97i·13-s + 2.65i·15-s − 2.65i·17-s + 8.26i·19-s − 5.23·21-s − 3.04·23-s + 25-s + 2.76i·27-s + 7.39i·29-s − 6.49·31-s + ⋯
L(s)  = 1  − 1.53i·3-s − 0.447·5-s − 0.745i·7-s − 1.34·9-s + 1.75i·11-s + 0.546i·13-s + 0.685i·15-s − 0.643i·17-s + 1.89i·19-s − 1.14·21-s − 0.634·23-s + 0.200·25-s + 0.532i·27-s + 1.37i·29-s − 1.16·31-s + ⋯

Functional equation

Λ(s)=(1640s/2ΓC(s)L(s)=((0.6720.740i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1640s/2ΓC(s+1/2)L(s)=((0.6720.740i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16401640    =    235412^{3} \cdot 5 \cdot 41
Sign: 0.6720.740i0.672 - 0.740i
Analytic conductor: 13.095413.0954
Root analytic conductor: 3.618763.61876
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1640(81,)\chi_{1640} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1640, ( :1/2), 0.6720.740i)(2,\ 1640,\ (\ :1/2),\ 0.672 - 0.740i)

Particular Values

L(1)L(1) \approx 0.76821684350.7682168435
L(12)L(\frac12) \approx 0.76821684350.7682168435
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+T 1 + T
41 1+(4.304.74i)T 1 + (4.30 - 4.74i)T
good3 1+2.65iT3T2 1 + 2.65iT - 3T^{2}
7 1+1.97iT7T2 1 + 1.97iT - 7T^{2}
11 15.81iT11T2 1 - 5.81iT - 11T^{2}
13 11.97iT13T2 1 - 1.97iT - 13T^{2}
17 1+2.65iT17T2 1 + 2.65iT - 17T^{2}
19 18.26iT19T2 1 - 8.26iT - 19T^{2}
23 1+3.04T+23T2 1 + 3.04T + 23T^{2}
29 17.39iT29T2 1 - 7.39iT - 29T^{2}
31 1+6.49T+31T2 1 + 6.49T + 31T^{2}
37 11.45T+37T2 1 - 1.45T + 37T^{2}
43 1+11.3T+43T2 1 + 11.3T + 43T^{2}
47 110.0iT47T2 1 - 10.0iT - 47T^{2}
53 1+9.33iT53T2 1 + 9.33iT - 53T^{2}
59 17.15T+59T2 1 - 7.15T + 59T^{2}
61 19.72T+61T2 1 - 9.72T + 61T^{2}
67 1+1.15iT67T2 1 + 1.15iT - 67T^{2}
71 1+4.05iT71T2 1 + 4.05iT - 71T^{2}
73 1+1.59T+73T2 1 + 1.59T + 73T^{2}
79 1+1.57iT79T2 1 + 1.57iT - 79T^{2}
83 14.11T+83T2 1 - 4.11T + 83T^{2}
89 117.9iT89T2 1 - 17.9iT - 89T^{2}
97 1+12.8iT97T2 1 + 12.8iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.558536054423451599104148525407, −8.290933198029373123355228148285, −7.78083026645325736212266234364, −6.93575743057629344395532924264, −6.82357110662940211354406547995, −5.51015308869037168052884429165, −4.45225240480219054903571784719, −3.49983361075254148535155330038, −2.03546466560503920474742035875, −1.40375150233782228300218548901, 0.29619079096306906098490376080, 2.56015149172486944881949137532, 3.46399976711725860526999046659, 4.09745345521367322470836044310, 5.26000144813132876462206314397, 5.64403730222312693147208662376, 6.72959813393347939876287851724, 8.071793308220773874806007203822, 8.687358924920554430164754236628, 9.139971828492265564348647188568

Graph of the ZZ-function along the critical line