↑

Properties

Label 2-333-37.6-c2-0-23
Degree $2$
Conductor $333$
Sign $-0.852 + 0.523i$
Analytic cond. $9.07359$
Root an. cond. $3.01224$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.123 − 0.123i)2-s + 3.96i·4-s + (−1.06 − 1.06i)5-s − 7.73·7-s + (0.984 + 0.984i)8-s − 0.264·10-s − 4.90i·11-s + (−0.397 − 0.397i)13-s + (−0.954 + 0.954i)14-s − 15.6·16-s + (−18.9 − 18.9i)17-s + (7.65 + 7.65i)19-s + (4.24 − 4.24i)20-s + (−0.605 − 0.605i)22-s + (−15.9 − 15.9i)23-s + ⋯
L(s)  = 1  + (0.0617 − 0.0617i)2-s + 0.992i·4-s + (−0.213 − 0.213i)5-s − 1.10·7-s + (0.123 + 0.123i)8-s − 0.0264·10-s − 0.445i·11-s + (−0.0305 − 0.0305i)13-s + (−0.0682 + 0.0682i)14-s − 0.977·16-s + (−1.11 − 1.11i)17-s + (0.403 + 0.403i)19-s + (0.212 − 0.212i)20-s + (−0.0275 − 0.0275i)22-s + (−0.695 − 0.695i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333\)    =    \(3^{2} \cdot 37\)
Sign: $-0.852 + 0.523i$
Analytic conductor: \(9.07359\)
Root analytic conductor: \(3.01224\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{333} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 333,\ (\ :1),\ -0.852 + 0.523i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0406947 - 0.144017i\)
\(L(\frac12)\) \(\approx\) \(0.0406947 - 0.144017i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 + (-36.5 - 5.60i)T \)
good2 \( 1 + (-0.123 + 0.123i)T - 4iT^{2} \)
5 \( 1 + (1.06 + 1.06i)T + 25iT^{2} \)
7 \( 1 + 7.73T + 49T^{2} \)
11 \( 1 + 4.90iT - 121T^{2} \)
13 \( 1 + (0.397 + 0.397i)T + 169iT^{2} \)
17 \( 1 + (18.9 + 18.9i)T + 289iT^{2} \)
19 \( 1 + (-7.65 - 7.65i)T + 361iT^{2} \)
23 \( 1 + (15.9 + 15.9i)T + 529iT^{2} \)
29 \( 1 + (15.6 - 15.6i)T - 841iT^{2} \)
31 \( 1 + (5.99 - 5.99i)T - 961iT^{2} \)
41 \( 1 + 2.08iT - 1.68e3T^{2} \)
43 \( 1 + (27.2 + 27.2i)T + 1.84e3iT^{2} \)
47 \( 1 + 56.3T + 2.20e3T^{2} \)
53 \( 1 + 83.7T + 2.80e3T^{2} \)
59 \( 1 + (-35.5 - 35.5i)T + 3.48e3iT^{2} \)
61 \( 1 + (34.2 - 34.2i)T - 3.72e3iT^{2} \)
67 \( 1 - 130. iT - 4.48e3T^{2} \)
71 \( 1 - 35.8T + 5.04e3T^{2} \)
73 \( 1 + 21.5iT - 5.32e3T^{2} \)
79 \( 1 + (-20.8 - 20.8i)T + 6.24e3iT^{2} \)
83 \( 1 - 93.4T + 6.88e3T^{2} \)
89 \( 1 + (25.0 - 25.0i)T - 7.92e3iT^{2} \)
97 \( 1 + (24.6 + 24.6i)T + 9.40e3iT^{2} \)
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

−11.18328867149167768355958099774, −9.938762076307366859486029582872, −9.019110383463125282107411152526, −8.187869772667761324228211263022, −7.11353046279500589527371234215, −6.25394311058578340436289079969, −4.70996543463529126126174651898, −3.60275400453142605911343696917, −2.59291741589870018969444311084, −0.06400226598292900245531743929, 1.89755339791365600119180226291, 3.52113316222141153469612779231, 4.78375808790713413183736605505, 6.07854190749893223858074014253, 6.65262672364427660272013604612, 7.84750188689827529941275265828, 9.354764379116546104247729303337, 9.702997259489644923606357614866, 10.81868859289170999903408141681, 11.50638909750158928699432037322

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