Properties

Label 2-46-23.16-c3-0-1
Degree $2$
Conductor $46$
Sign $0.0300 + 0.999i$
Analytic cond. $2.71408$
Root an. cond. $1.64744$
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  + (−1.30 + 1.51i)2-s + (−4.75 + 3.05i)3-s + (−0.569 − 3.95i)4-s + (−6.73 − 14.7i)5-s + (1.60 − 11.1i)6-s + (22.9 − 6.72i)7-s + (6.73 + 4.32i)8-s + (2.04 − 4.48i)9-s + (31.1 + 9.13i)10-s + (−43.2 − 49.9i)11-s + (14.8 + 17.0i)12-s + (−66.3 − 19.4i)13-s + (−19.8 + 43.4i)14-s + (77.1 + 49.5i)15-s + (−15.3 + 4.50i)16-s + (2.87 − 20.0i)17-s + ⋯
L(s)  = 1  + (−0.463 + 0.534i)2-s + (−0.914 + 0.587i)3-s + (−0.0711 − 0.494i)4-s + (−0.602 − 1.31i)5-s + (0.109 − 0.761i)6-s + (1.23 − 0.363i)7-s + (0.297 + 0.191i)8-s + (0.0758 − 0.166i)9-s + (0.984 + 0.288i)10-s + (−1.18 − 1.36i)11-s + (0.356 + 0.410i)12-s + (−1.41 − 0.415i)13-s + (−0.378 + 0.829i)14-s + (1.32 + 0.852i)15-s + (−0.239 + 0.0704i)16-s + (0.0410 − 0.285i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0300 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0300 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46\)    =    \(2 \cdot 23\)
Sign: $0.0300 + 0.999i$
Analytic conductor: \(2.71408\)
Root analytic conductor: \(1.64744\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{46} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 46,\ (\ :3/2),\ 0.0300 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.316830 - 0.307443i\)
\(L(\frac12)\) \(\approx\) \(0.316830 - 0.307443i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 1.51i)T \)
23 \( 1 + (-68.6 + 86.3i)T \)
good3 \( 1 + (4.75 - 3.05i)T + (11.2 - 24.5i)T^{2} \)
5 \( 1 + (6.73 + 14.7i)T + (-81.8 + 94.4i)T^{2} \)
7 \( 1 + (-22.9 + 6.72i)T + (288. - 185. i)T^{2} \)
11 \( 1 + (43.2 + 49.9i)T + (-189. + 1.31e3i)T^{2} \)
13 \( 1 + (66.3 + 19.4i)T + (1.84e3 + 1.18e3i)T^{2} \)
17 \( 1 + (-2.87 + 20.0i)T + (-4.71e3 - 1.38e3i)T^{2} \)
19 \( 1 + (-9.51 - 66.1i)T + (-6.58e3 + 1.93e3i)T^{2} \)
29 \( 1 + (21.4 - 149. i)T + (-2.34e4 - 6.87e3i)T^{2} \)
31 \( 1 + (-55.7 - 35.8i)T + (1.23e4 + 2.70e4i)T^{2} \)
37 \( 1 + (46.8 - 102. i)T + (-3.31e4 - 3.82e4i)T^{2} \)
41 \( 1 + (105. + 230. i)T + (-4.51e4 + 5.20e4i)T^{2} \)
43 \( 1 + (-221. + 142. i)T + (3.30e4 - 7.23e4i)T^{2} \)
47 \( 1 + 290.T + 1.03e5T^{2} \)
53 \( 1 + (34.6 - 10.1i)T + (1.25e5 - 8.04e4i)T^{2} \)
59 \( 1 + (-110. - 32.5i)T + (1.72e5 + 1.11e5i)T^{2} \)
61 \( 1 + (-230. - 148. i)T + (9.42e4 + 2.06e5i)T^{2} \)
67 \( 1 + (-476. + 550. i)T + (-4.28e4 - 2.97e5i)T^{2} \)
71 \( 1 + (-116. + 134. i)T + (-5.09e4 - 3.54e5i)T^{2} \)
73 \( 1 + (89.6 + 623. i)T + (-3.73e5 + 1.09e5i)T^{2} \)
79 \( 1 + (-636. - 186. i)T + (4.14e5 + 2.66e5i)T^{2} \)
83 \( 1 + (33.4 - 73.1i)T + (-3.74e5 - 4.32e5i)T^{2} \)
89 \( 1 + (704. - 452. i)T + (2.92e5 - 6.41e5i)T^{2} \)
97 \( 1 + (533. + 1.16e3i)T + (-5.97e5 + 6.89e5i)T^{2} \)
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

−15.43660851400682076728669751686, −14.05428985025145771477455338354, −12.45915857795366815119935472339, −11.23643187732995020873153119636, −10.32224268307751011261921736463, −8.540149987674828164290598595359, −7.76202137822028028750697109046, −5.32382357949041402824691206794, −4.85680711756941711112055199798, −0.43856390928630386584503191069, 2.36334347254133467450214743501, 4.93171914893608530673872369116, 7.04222547187434132091737666018, 7.74783280751103645433963712465, 9.878775151286544845102491812743, 11.17393353693859695441921296084, 11.67372085352648130685755894220, 12.77676503902424399275192819159, 14.65298331715128006312773116876, 15.36436027363582814131791043778

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