↑

Properties

Label 2-2268-9.4-c1-0-14
Degree 22
Conductor 22682268
Sign 0.173+0.984i0.173 + 0.984i
Analytic cond. 18.110018.1100
Root an. cond. 4.255594.25559
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  + (−2 + 3.46i)5-s + (0.5 + 0.866i)7-s + (−1 − 1.73i)11-s + (3 − 5.19i)13-s − 4·17-s − 4·19-s + (−1 + 1.73i)23-s + (−5.49 − 9.52i)25-s + (1 + 1.73i)29-s − 3.99·35-s + 2·37-s + (2 + 3.46i)43-s + (−6 − 10.3i)47-s + (−0.499 + 0.866i)49-s − 6·53-s + ⋯
L(s)  = 1  + (−0.894 + 1.54i)5-s + (0.188 + 0.327i)7-s + (−0.301 − 0.522i)11-s + (0.832 − 1.44i)13-s − 0.970·17-s − 0.917·19-s + (−0.208 + 0.361i)23-s + (−1.09 − 1.90i)25-s + (0.185 + 0.321i)29-s − 0.676·35-s + 0.328·37-s + (0.304 + 0.528i)43-s + (−0.875 − 1.51i)47-s + (−0.0714 + 0.123i)49-s − 0.824·53-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=((0.173+0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s+1/2) L(s)=((0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: 0.173+0.984i0.173 + 0.984i
Analytic conductor: 18.110018.1100
Root analytic conductor: 4.255594.25559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2268(757,⋅)\chi_{2268} (757, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2268, ( :1/2), 0.173+0.984i)(2,\ 2268,\ (\ :1/2),\ 0.173 + 0.984i)

Particular Values

L(1)L(1) ≈\approx 0.64830798350.6483079835
L(12)L(\frac12) ≈\approx 0.64830798350.6483079835
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
7 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good5 1+(2−3.46i)T+(−2.5−4.33i)T2 1 + (2 - 3.46i)T + (-2.5 - 4.33i)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+4T+17T2 1 + 4T + 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+(1−1.73i)T+(−11.5−19.9i)T2 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1−1.73i)T+(−14.5+25.1i)T2 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2}
31 1+(−15.5−26.8i)T2 1 + (-15.5 - 26.8i)T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+(−20.5−35.5i)T2 1 + (-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+(6+10.3i)T+(−23.5+40.7i)T2 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1+(−4+6.92i)T+(−29.5−51.0i)T2 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2}
61 1+(3+5.19i)T+(−30.5+52.8i)T2 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2}
67 1+(−4+6.92i)T+(−33.5−58.0i)T2 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2}
71 1−14T+71T2 1 - 14T + 71T^{2}
73 1+2T+73T2 1 + 2T + 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+(−2−3.46i)T+(−41.5+71.8i)T2 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2}
89 1+89T2 1 + 89T^{2}
97 1+(−1−1.73i)T+(−48.5+84.0i)T2 1 + (-1 - 1.73i)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

−8.452445320215308997712389181311, −8.205117874874520281137929420529, −7.33160728326287922592009280461, −6.48540086436010212859951352585, −5.94992491857325362193599234821, −4.80409568455309079988321728165, −3.65201360089004495319850394693, −3.15863311698894899894797157765, −2.17742265112025065238118950789, −0.24271602992348347599577944285, 1.15292812845011734846901312917, 2.19041690873596747963118378836, 3.90156045537588755392256473343, 4.35650923732835223176343893425, 4.89126820607902665139450353310, 6.10929636377027391011311032325, 6.92145705418427138185999443698, 7.83623571249929562897794842347, 8.513209991261983443747381470336, 8.975304893681558232217073111616

Graph of the ZZ-function along the critical line