Properties

Label 2-23-23.16-c3-0-3
Degree $2$
Conductor $23$
Sign $0.746 + 0.665i$
Analytic cond. $1.35704$
Root an. cond. $1.16492$
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  + (1.61 − 1.86i)2-s + (1.97 − 1.26i)3-s + (0.275 + 1.91i)4-s + (−3.07 − 6.73i)5-s + (0.822 − 5.71i)6-s + (−11.6 + 3.43i)7-s + (20.5 + 13.2i)8-s + (−8.92 + 19.5i)9-s + (−17.4 − 5.13i)10-s + (3.34 + 3.85i)11-s + (2.97 + 3.43i)12-s + (−23.2 − 6.83i)13-s + (−12.4 + 27.2i)14-s + (−14.6 − 9.39i)15-s + (42.9 − 12.6i)16-s + (9.15 − 63.6i)17-s + ⋯
L(s)  = 1  + (0.570 − 0.657i)2-s + (0.379 − 0.244i)3-s + (0.0344 + 0.239i)4-s + (−0.275 − 0.602i)5-s + (0.0559 − 0.389i)6-s + (−0.631 + 0.185i)7-s + (0.909 + 0.584i)8-s + (−0.330 + 0.724i)9-s + (−0.553 − 0.162i)10-s + (0.0916 + 0.105i)11-s + (0.0716 + 0.0826i)12-s + (−0.496 − 0.145i)13-s + (−0.237 + 0.521i)14-s + (−0.251 − 0.161i)15-s + (0.670 − 0.197i)16-s + (0.130 − 0.908i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.746 + 0.665i$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ 0.746 + 0.665i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.39952 - 0.533774i\)
\(L(\frac12)\) \(\approx\) \(1.39952 - 0.533774i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-72.4 + 83.1i)T \)
good2 \( 1 + (-1.61 + 1.86i)T + (-1.13 - 7.91i)T^{2} \)
3 \( 1 + (-1.97 + 1.26i)T + (11.2 - 24.5i)T^{2} \)
5 \( 1 + (3.07 + 6.73i)T + (-81.8 + 94.4i)T^{2} \)
7 \( 1 + (11.6 - 3.43i)T + (288. - 185. i)T^{2} \)
11 \( 1 + (-3.34 - 3.85i)T + (-189. + 1.31e3i)T^{2} \)
13 \( 1 + (23.2 + 6.83i)T + (1.84e3 + 1.18e3i)T^{2} \)
17 \( 1 + (-9.15 + 63.6i)T + (-4.71e3 - 1.38e3i)T^{2} \)
19 \( 1 + (7.05 + 49.0i)T + (-6.58e3 + 1.93e3i)T^{2} \)
29 \( 1 + (24.5 - 170. i)T + (-2.34e4 - 6.87e3i)T^{2} \)
31 \( 1 + (195. + 125. i)T + (1.23e4 + 2.70e4i)T^{2} \)
37 \( 1 + (98.8 - 216. i)T + (-3.31e4 - 3.82e4i)T^{2} \)
41 \( 1 + (-129. - 283. i)T + (-4.51e4 + 5.20e4i)T^{2} \)
43 \( 1 + (-442. + 284. i)T + (3.30e4 - 7.23e4i)T^{2} \)
47 \( 1 - 407.T + 1.03e5T^{2} \)
53 \( 1 + (461. - 135. i)T + (1.25e5 - 8.04e4i)T^{2} \)
59 \( 1 + (-41.1 - 12.0i)T + (1.72e5 + 1.11e5i)T^{2} \)
61 \( 1 + (286. + 183. i)T + (9.42e4 + 2.06e5i)T^{2} \)
67 \( 1 + (347. - 400. i)T + (-4.28e4 - 2.97e5i)T^{2} \)
71 \( 1 + (-488. + 563. i)T + (-5.09e4 - 3.54e5i)T^{2} \)
73 \( 1 + (-43.0 - 299. i)T + (-3.73e5 + 1.09e5i)T^{2} \)
79 \( 1 + (744. + 218. i)T + (4.14e5 + 2.66e5i)T^{2} \)
83 \( 1 + (-121. + 266. i)T + (-3.74e5 - 4.32e5i)T^{2} \)
89 \( 1 + (-479. + 307. i)T + (2.92e5 - 6.41e5i)T^{2} \)
97 \( 1 + (-776. - 1.70e3i)T + (-5.97e5 + 6.89e5i)T^{2} \)
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

−17.02806826603081328108874359777, −16.11704702488229372262985398107, −14.31943540474803419145332088585, −13.10612674973092973793398111796, −12.32855101348388968192382960198, −10.92109110003083028746170990696, −8.953509415801991109887457010653, −7.46492861007565883041476240731, −4.84370593803020163814755552561, −2.78500731338958596124620794463, 3.72785995946286649433956837095, 5.95505189693845551092908837608, 7.30371093670489269138934580083, 9.407029431723990040804514585269, 10.79402099183297901768161297915, 12.66016004852257040562165520870, 14.18230859175537743267135965748, 14.90652784503063679155362476609, 15.87983503760406597634044247633, 17.24604963376440013166099757606

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