↑

Properties

Label 2-297-9.4-c1-0-3
Degree $2$
Conductor $297$
Sign $0.173 - 0.984i$
Analytic cond. $2.37155$
Root an. cond. $1.53998$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41991 + 1.19144i\)
\(L(\frac12)\) \(\approx\) \(1.41991 + 1.19144i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line