↑

Properties

Label 2-297-9.4-c1-0-3
Degree 22
Conductor 297297
Sign 0.173−0.984i0.173 - 0.984i
Analytic cond. 2.371552.37155
Root an. cond. 1.539981.53998
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  + (0.766 + 1.32i)2-s + (−0.173 + 0.300i)4-s + (−0.266 + 0.460i)5-s + (1.43 + 2.49i)7-s + 2.53·8-s − 0.815·10-s + (−0.5 − 0.866i)11-s + (−0.0320 + 0.0555i)13-s + (−2.20 + 3.82i)14-s + (2.28 + 3.96i)16-s − 1.22·17-s + 0.411·19-s + (−0.0923 − 0.160i)20-s + (0.766 − 1.32i)22-s + (−2.11 + 3.66i)23-s + ⋯
L(s)  = 1  + (0.541 + 0.938i)2-s + (−0.0868 + 0.150i)4-s + (−0.118 + 0.206i)5-s + (0.544 + 0.942i)7-s + 0.895·8-s − 0.257·10-s + (−0.150 − 0.261i)11-s + (−0.00889 + 0.0154i)13-s + (−0.589 + 1.02i)14-s + (0.571 + 0.990i)16-s − 0.297·17-s + 0.0943·19-s + (−0.0206 − 0.0357i)20-s + (0.163 − 0.282i)22-s + (−0.440 + 0.763i)23-s + ⋯

Functional equation

Λ(s)=(297s/2 ΓC(s) L(s)=((0.173−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(297s/2 ΓC(s+1/2) L(s)=((0.173−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{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: 297297    =    33⋅113^{3} \cdot 11
Sign: 0.173−0.984i0.173 - 0.984i
Analytic conductor: 2.371552.37155
Root analytic conductor: 1.539981.53998
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ297(199,⋅)\chi_{297} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 297, ( :1/2), 0.173−0.984i)(2,\ 297,\ (\ :1/2),\ 0.173 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 1.41991+1.19144i1.41991 + 1.19144i
L(12)L(\frac12) ≈\approx 1.41991+1.19144i1.41991 + 1.19144i
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)
bad3 1 1
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good2 1+(−0.766−1.32i)T+(−1+1.73i)T2 1 + (-0.766 - 1.32i)T + (-1 + 1.73i)T^{2}
5 1+(0.266−0.460i)T+(−2.5−4.33i)T2 1 + (0.266 - 0.460i)T + (-2.5 - 4.33i)T^{2}
7 1+(−1.43−2.49i)T+(−3.5+6.06i)T2 1 + (-1.43 - 2.49i)T + (-3.5 + 6.06i)T^{2}
13 1+(0.0320−0.0555i)T+(−6.5−11.2i)T2 1 + (0.0320 - 0.0555i)T + (-6.5 - 11.2i)T^{2}
17 1+1.22T+17T2 1 + 1.22T + 17T^{2}
19 1−0.411T+19T2 1 - 0.411T + 19T^{2}
23 1+(2.11−3.66i)T+(−11.5−19.9i)T2 1 + (2.11 - 3.66i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.16+7.21i)T+(−14.5+25.1i)T2 1 + (4.16 + 7.21i)T + (-14.5 + 25.1i)T^{2}
31 1+(−3.97+6.87i)T+(−15.5−26.8i)T2 1 + (-3.97 + 6.87i)T + (-15.5 - 26.8i)T^{2}
37 1+8.94T+37T2 1 + 8.94T + 37T^{2}
41 1+(−4.46+7.73i)T+(−20.5−35.5i)T2 1 + (-4.46 + 7.73i)T + (-20.5 - 35.5i)T^{2}
43 1+(5.71+9.90i)T+(−21.5+37.2i)T2 1 + (5.71 + 9.90i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.92−3.34i)T+(−23.5+40.7i)T2 1 + (-1.92 - 3.34i)T + (-23.5 + 40.7i)T^{2}
53 1+0.448T+53T2 1 + 0.448T + 53T^{2}
59 1+(−6.84+11.8i)T+(−29.5−51.0i)T2 1 + (-6.84 + 11.8i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.56−4.43i)T+(−30.5+52.8i)T2 1 + (-2.56 - 4.43i)T + (-30.5 + 52.8i)T^{2}
67 1+(4.92−8.52i)T+(−33.5−58.0i)T2 1 + (4.92 - 8.52i)T + (-33.5 - 58.0i)T^{2}
71 1+4.49T+71T2 1 + 4.49T + 71T^{2}
73 1+8.96T+73T2 1 + 8.96T + 73T^{2}
79 1+(−1.32−2.29i)T+(−39.5+68.4i)T2 1 + (-1.32 - 2.29i)T + (-39.5 + 68.4i)T^{2}
83 1+(−2.81−4.88i)T+(−41.5+71.8i)T2 1 + (-2.81 - 4.88i)T + (-41.5 + 71.8i)T^{2}
89 1+7.97T+89T2 1 + 7.97T + 89T^{2}
97 1+(−4.95−8.57i)T+(−48.5+84.0i)T2 1 + (-4.95 - 8.57i)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

−11.91215441701222008215106122830, −11.20347483502850657081615142587, −10.11211667131774949839845447804, −8.887656582951488025830620453225, −7.909542016540627396988374135642, −7.02911022623546907566121726890, −5.83925959937992546147708300809, −5.26960797486394717916836218389, −3.92940872116791623355149130334, −2.10604677286554140475546673027, 1.47591518981389844238561761748, 3.01656996969358622702331947137, 4.26367213984918615633490373655, 4.95785147103169844779059838351, 6.74498004204457063246582503206, 7.68897565613809539555358448079, 8.684295828914646412947464059101, 10.21090211165993046843035163177, 10.66939620380039898416296392773, 11.61822489770083628129577485855

Graph of the ZZ-function along the critical line