↑

Properties

Label 2-1332-37.12-c1-0-7
Degree 22
Conductor 13321332
Sign 0.766−0.641i0.766 - 0.641i
Analytic cond. 10.636010.6360
Root an. cond. 3.261293.26129
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.68 + 0.613i)5-s + (2.82 − 1.02i)7-s + (1.22 + 2.12i)11-s + (−0.123 + 0.698i)13-s + (0.266 + 1.51i)17-s + (−0.404 − 0.339i)19-s + (0.594 − 1.02i)23-s + (−1.36 + 1.14i)25-s + (0.173 + 0.300i)29-s + 7.53·31-s + (−4.13 + 3.46i)35-s + (−0.665 + 6.04i)37-s + (1.20 − 6.86i)41-s + 3.76·43-s + (−1.56 + 2.70i)47-s + ⋯
L(s)  = 1  + (−0.753 + 0.274i)5-s + (1.06 − 0.388i)7-s + (0.369 + 0.640i)11-s + (−0.0341 + 0.193i)13-s + (0.0647 + 0.366i)17-s + (−0.0928 − 0.0779i)19-s + (0.123 − 0.214i)23-s + (−0.273 + 0.229i)25-s + (0.0322 + 0.0558i)29-s + 1.35·31-s + (−0.698 + 0.586i)35-s + (−0.109 + 0.993i)37-s + (0.188 − 1.07i)41-s + 0.574·43-s + (−0.227 + 0.394i)47-s + ⋯

Functional equation

Λ(s)=(1332s/2 ΓC(s) L(s)=((0.766−0.641i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1332s/2 ΓC(s+1/2) L(s)=((0.766−0.641i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13321332    =    22⋅32⋅372^{2} \cdot 3^{2} \cdot 37
Sign: 0.766−0.641i0.766 - 0.641i
Analytic conductor: 10.636010.6360
Root analytic conductor: 3.261293.26129
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1332(937,⋅)\chi_{1332} (937, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1332, ( :1/2), 0.766−0.641i)(2,\ 1332,\ (\ :1/2),\ 0.766 - 0.641i)

Particular Values

L(1)L(1) ≈\approx 1.6052773461.605277346
L(12)L(\frac12) ≈\approx 1.6052773461.605277346
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
3 1 1
37 1+(0.665−6.04i)T 1 + (0.665 - 6.04i)T
good5 1+(1.68−0.613i)T+(3.83−3.21i)T2 1 + (1.68 - 0.613i)T + (3.83 - 3.21i)T^{2}
7 1+(−2.82+1.02i)T+(5.36−4.49i)T2 1 + (-2.82 + 1.02i)T + (5.36 - 4.49i)T^{2}
11 1+(−1.22−2.12i)T+(−5.5+9.52i)T2 1 + (-1.22 - 2.12i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.123−0.698i)T+(−12.2−4.44i)T2 1 + (0.123 - 0.698i)T + (-12.2 - 4.44i)T^{2}
17 1+(−0.266−1.51i)T+(−15.9+5.81i)T2 1 + (-0.266 - 1.51i)T + (-15.9 + 5.81i)T^{2}
19 1+(0.404+0.339i)T+(3.29+18.7i)T2 1 + (0.404 + 0.339i)T + (3.29 + 18.7i)T^{2}
23 1+(−0.594+1.02i)T+(−11.5−19.9i)T2 1 + (-0.594 + 1.02i)T + (-11.5 - 19.9i)T^{2}
29 1+(−0.173−0.300i)T+(−14.5+25.1i)T2 1 + (-0.173 - 0.300i)T + (-14.5 + 25.1i)T^{2}
31 1−7.53T+31T2 1 - 7.53T + 31T^{2}
41 1+(−1.20+6.86i)T+(−38.5−14.0i)T2 1 + (-1.20 + 6.86i)T + (-38.5 - 14.0i)T^{2}
43 1−3.76T+43T2 1 - 3.76T + 43T^{2}
47 1+(1.56−2.70i)T+(−23.5−40.7i)T2 1 + (1.56 - 2.70i)T + (-23.5 - 40.7i)T^{2}
53 1+(−9.80−3.56i)T+(40.6+34.0i)T2 1 + (-9.80 - 3.56i)T + (40.6 + 34.0i)T^{2}
59 1+(−10.8−3.93i)T+(45.1+37.9i)T2 1 + (-10.8 - 3.93i)T + (45.1 + 37.9i)T^{2}
61 1+(1.34−7.64i)T+(−57.3−20.8i)T2 1 + (1.34 - 7.64i)T + (-57.3 - 20.8i)T^{2}
67 1+(−5.01+1.82i)T+(51.3−43.0i)T2 1 + (-5.01 + 1.82i)T + (51.3 - 43.0i)T^{2}
71 1+(−4.62−3.87i)T+(12.3+69.9i)T2 1 + (-4.62 - 3.87i)T + (12.3 + 69.9i)T^{2}
73 1−6.37T+73T2 1 - 6.37T + 73T^{2}
79 1+(11.3−4.11i)T+(60.5−50.7i)T2 1 + (11.3 - 4.11i)T + (60.5 - 50.7i)T^{2}
83 1+(−1.94−11.0i)T+(−77.9+28.3i)T2 1 + (-1.94 - 11.0i)T + (-77.9 + 28.3i)T^{2}
89 1+(−2.57−0.936i)T+(68.1+57.2i)T2 1 + (-2.57 - 0.936i)T + (68.1 + 57.2i)T^{2}
97 1+(1.94−3.36i)T+(−48.5−84.0i)T2 1 + (1.94 - 3.36i)T + (-48.5 - 84.0i)T^{2}
show more
show less
   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

−9.809231632901829585106622961408, −8.722126161170743724457022697607, −8.058437688061811963468128999898, −7.33007259941779734727892096154, −6.63504491415184805045867170589, −5.41324050118075165348727948717, −4.42773446136772462027903778889, −3.88370685571508670339985386621, −2.48504313888771088549759841384, −1.19939663238097430398270335919, 0.802461304236057234486921536106, 2.22751085010694853391328842050, 3.50201475358799647306637396638, 4.44186552524568338678592133432, 5.23290456354233684500240285057, 6.16378982712918265071084571479, 7.23909230988294545282934577657, 8.155462460394806324484342901322, 8.444667224781143684999848407161, 9.432198842815734896876883951978

Graph of the ZZ-function along the critical line