↑

Properties

Label 2-1350-9.4-c1-0-8
Degree 22
Conductor 13501350
Sign 0.766−0.642i0.766 - 0.642i
Analytic cond. 10.779810.7798
Root an. cond. 3.283263.28326
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)7-s − 0.999·8-s + (−1 − 1.73i)11-s + (3 − 5.19i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 2·17-s + 6·19-s + (0.999 − 1.73i)22-s + (0.5 − 0.866i)23-s + 6·26-s − 0.999·28-s + (4.5 + 7.79i)29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.188 + 0.327i)7-s − 0.353·8-s + (−0.301 − 0.522i)11-s + (0.832 − 1.44i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.485·17-s + 1.37·19-s + (0.213 − 0.369i)22-s + (0.104 − 0.180i)23-s + 1.17·26-s − 0.188·28-s + (0.835 + 1.44i)29-s + ⋯

Functional equation

Λ(s)=(1350s/2 ΓC(s) L(s)=((0.766−0.642i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1350s/2 ΓC(s+1/2) L(s)=((0.766−0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13501350    =    2⋅33⋅522 \cdot 3^{3} \cdot 5^{2}
Sign: 0.766−0.642i0.766 - 0.642i
Analytic conductor: 10.779810.7798
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1350(901,⋅)\chi_{1350} (901, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1350, ( :1/2), 0.766−0.642i)(2,\ 1350,\ (\ :1/2),\ 0.766 - 0.642i)

Particular Values

L(1)L(1) ≈\approx 2.1020549662.102054966
L(12)L(\frac12) ≈\approx 2.1020549662.102054966
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
3 1 1
5 1 1
good7 1+(−0.5−0.866i)T+(−3.5+6.06i)T2 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2}
11 1+(1+1.73i)T+(−5.5+9.52i)T2 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}
13 1+(−3+5.19i)T+(−6.5−11.2i)T2 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
23 1+(−0.5+0.866i)T+(−11.5−19.9i)T2 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2}
29 1+(−4.5−7.79i)T+(−14.5+25.1i)T2 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1+1.73i)T+(−15.5−26.8i)T2 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+(5.5−9.52i)T+(−20.5−35.5i)T2 1 + (5.5 - 9.52i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2−3.46i)T+(−21.5+37.2i)T2 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2}
47 1+(3.5+6.06i)T+(−23.5+40.7i)T2 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2}
53 1+53T2 1 + 53T^{2}
59 1+(2−3.46i)T+(−29.5−51.0i)T2 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.5−6.06i)T+(−30.5+52.8i)T2 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.5+9.52i)T+(−33.5−58.0i)T2 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2}
71 1−6T+71T2 1 - 6T + 71T^{2}
73 1+4T+73T2 1 + 4T + 73T^{2}
79 1+(−6−10.3i)T+(−39.5+68.4i)T2 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2}
83 1+(5.5+9.52i)T+(−41.5+71.8i)T2 1 + (5.5 + 9.52i)T + (-41.5 + 71.8i)T^{2}
89 1+T+89T2 1 + T + 89T^{2}
97 1+(−4−6.92i)T+(−48.5+84.0i)T2 1 + (-4 - 6.92i)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

−9.652709068434193011728793665670, −8.534239114370644041235166018900, −8.153927211192121552050246523868, −7.28249958751412230256077157485, −6.26795048789240915354021246679, −5.49168491834556954928837097563, −4.94421653299149623475874815360, −3.49539953319052853498205150878, −2.93027409699647923000437327128, −1.03038065239971203891007142409, 1.11919988156753806813512814014, 2.23833209356571198125667153809, 3.46938548816308408760051880023, 4.28798078596763913223665098551, 5.13509841862378702042054862255, 6.12705299990550386674174035839, 7.03641321811563206808671022494, 7.899356743076805658758081521360, 8.904348762536853763362943191910, 9.661551721970156061428062779576

Graph of the ZZ-function along the critical line