Properties

Label 2-33-33.17-c5-0-7
Degree $2$
Conductor $33$
Sign $0.420 - 0.907i$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.489 + 1.50i)2-s + (4.79 + 14.8i)3-s + (23.8 − 17.3i)4-s + (16.6 + 5.40i)5-s + (−20.0 + 14.5i)6-s + (36.5 + 50.2i)7-s + (78.8 + 57.3i)8-s + (−196. + 142. i)9-s + 27.7i·10-s + (−56.2 + 397. i)11-s + (371. + 270. i)12-s + (283. − 92.1i)13-s + (−57.9 + 79.7i)14-s + (−0.335 + 272. i)15-s + (243. − 750. i)16-s + (320. − 985. i)17-s + ⋯
L(s)  = 1  + (0.0866 + 0.266i)2-s + (0.307 + 0.951i)3-s + (0.745 − 0.541i)4-s + (0.297 + 0.0967i)5-s + (−0.226 + 0.164i)6-s + (0.281 + 0.387i)7-s + (0.435 + 0.316i)8-s + (−0.810 + 0.585i)9-s + 0.0877i·10-s + (−0.140 + 0.990i)11-s + (0.744 + 0.542i)12-s + (0.465 − 0.151i)13-s + (−0.0790 + 0.108i)14-s + (−0.000385 + 0.313i)15-s + (0.238 − 0.732i)16-s + (0.268 − 0.826i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.420 - 0.907i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 0.420 - 0.907i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.79338 + 1.14535i\)
\(L(\frac12)\) \(\approx\) \(1.79338 + 1.14535i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.79 - 14.8i)T \)
11 \( 1 + (56.2 - 397. i)T \)
good2 \( 1 + (-0.489 - 1.50i)T + (-25.8 + 18.8i)T^{2} \)
5 \( 1 + (-16.6 - 5.40i)T + (2.52e3 + 1.83e3i)T^{2} \)
7 \( 1 + (-36.5 - 50.2i)T + (-5.19e3 + 1.59e4i)T^{2} \)
13 \( 1 + (-283. + 92.1i)T + (3.00e5 - 2.18e5i)T^{2} \)
17 \( 1 + (-320. + 985. i)T + (-1.14e6 - 8.34e5i)T^{2} \)
19 \( 1 + (821. - 1.13e3i)T + (-7.65e5 - 2.35e6i)T^{2} \)
23 \( 1 + 3.74e3iT - 6.43e6T^{2} \)
29 \( 1 + (-5.35e3 + 3.88e3i)T + (6.33e6 - 1.95e7i)T^{2} \)
31 \( 1 + (153. + 473. i)T + (-2.31e7 + 1.68e7i)T^{2} \)
37 \( 1 + (-219. + 159. i)T + (2.14e7 - 6.59e7i)T^{2} \)
41 \( 1 + (3.71e3 + 2.69e3i)T + (3.58e7 + 1.10e8i)T^{2} \)
43 \( 1 - 1.41e4iT - 1.47e8T^{2} \)
47 \( 1 + (2.15e3 - 2.96e3i)T + (-7.08e7 - 2.18e8i)T^{2} \)
53 \( 1 + (1.25e4 - 4.07e3i)T + (3.38e8 - 2.45e8i)T^{2} \)
59 \( 1 + (5.22e3 + 7.19e3i)T + (-2.20e8 + 6.79e8i)T^{2} \)
61 \( 1 + (2.29e4 + 7.46e3i)T + (6.83e8 + 4.96e8i)T^{2} \)
67 \( 1 + 5.39e4T + 1.35e9T^{2} \)
71 \( 1 + (-9.42e3 - 3.06e3i)T + (1.45e9 + 1.06e9i)T^{2} \)
73 \( 1 + (-3.52e3 - 4.85e3i)T + (-6.40e8 + 1.97e9i)T^{2} \)
79 \( 1 + (-9.39e4 + 3.05e4i)T + (2.48e9 - 1.80e9i)T^{2} \)
83 \( 1 + (3.57e4 - 1.09e5i)T + (-3.18e9 - 2.31e9i)T^{2} \)
89 \( 1 - 1.13e5iT - 5.58e9T^{2} \)
97 \( 1 + (-5.21e3 - 1.60e4i)T + (-6.94e9 + 5.04e9i)T^{2} \)
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

−15.72880844935782172888273334150, −14.88163060998590816063756410471, −13.99428572811074198712104482259, −12.01811638805315986200434517495, −10.64493264826964078333494820167, −9.758157058070178923225289709763, −8.076125561579450929263787835360, −6.21246574805974244668375548813, −4.74524061250738295482723037538, −2.38458277359853747334573840125, 1.53523886583981616768760188356, 3.32997006612416403977047121857, 6.12354130898425926886410117017, 7.48303765824400878687239891378, 8.661603289341752508579161654842, 10.78817673701791957081261648757, 11.82302703567375605982801786497, 13.09281191689076874089413373471, 13.87538290261889169947969674827, 15.47434336709515353089736171108

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