↑

Properties

Label 2-135-9.7-c3-0-11
Degree $2$
Conductor $135$
Sign $-0.847 - 0.530i$
Analytic cond. $7.96525$
Root an. cond. $2.82227$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.69 − 4.66i)2-s + (−10.4 − 18.1i)4-s + (−2.5 − 4.33i)5-s + (−6.28 + 10.8i)7-s − 69.9·8-s − 26.9·10-s + (6.41 − 11.1i)11-s + (−29.9 − 51.8i)13-s + (33.8 + 58.6i)14-s + (−104. + 180. i)16-s + 110.·17-s − 12.0·19-s + (−52.4 + 90.9i)20-s + (−34.5 − 59.8i)22-s + (−33.8 − 58.6i)23-s + ⋯
L(s)  = 1  + (0.951 − 1.64i)2-s + (−1.31 − 2.27i)4-s + (−0.223 − 0.387i)5-s + (−0.339 + 0.587i)7-s − 3.09·8-s − 0.851·10-s + (0.175 − 0.304i)11-s + (−0.638 − 1.10i)13-s + (0.646 + 1.11i)14-s + (−1.63 + 2.82i)16-s + 1.56·17-s − 0.145·19-s + (−0.586 + 1.01i)20-s + (−0.334 − 0.579i)22-s + (−0.306 − 0.531i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.847 - 0.530i$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ -0.847 - 0.530i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.512381 + 1.78593i\)
\(L(\frac12)\) \(\approx\) \(0.512381 + 1.78593i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.5 + 4.33i)T \)
good2 \( 1 + (-2.69 + 4.66i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (6.28 - 10.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-6.41 + 11.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (29.9 + 51.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 110.T + 4.91e3T^{2} \)
19 \( 1 + 12.0T + 6.85e3T^{2} \)
23 \( 1 + (33.8 + 58.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-99.9 + 173. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (38.3 + 66.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 22.4T + 5.06e4T^{2} \)
41 \( 1 + (-43.8 - 76.0i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (59.7 - 103. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-121. + 210. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 293.T + 1.48e5T^{2} \)
59 \( 1 + (290. + 503. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-386. + 670. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-115. - 200. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 744.T + 3.57e5T^{2} \)
73 \( 1 + 264.T + 3.89e5T^{2} \)
79 \( 1 + (279. - 484. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (610. - 1.05e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 255.T + 7.04e5T^{2} \)
97 \( 1 + (524. - 908. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−12.33863096968885131895384813059, −11.43215770402906286922823399860, −10.23328098862209776518947611638, −9.574224451972542077843849521911, −8.188468353030854207188122358875, −5.95008697814143867278776730469, −5.06057805691969241281675694176, −3.66512750424910979823836664849, −2.53403477169320153610324692242, −0.71009730850721411325575145521, 3.38955880579021272676939640927, 4.47024299298095498654130237909, 5.73514716889581418181621315233, 6.99014105372836144749677474741, 7.42393922527148934494681270949, 8.773494416363395922465802663034, 10.05036745956710823346932650679, 11.85956080725380565510374622862, 12.59086514068882432845115219204, 13.83380875791400150655365577785

Graph of the $Z$-function along the critical line