↑

Properties

Label 1-392-392.61-r1-0-0
Degree $1$
Conductor $392$
Sign $0.725 - 0.687i$
Analytic cond. $42.1262$
Root an. cond. $42.1262$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0747 − 0.997i)3-s + (0.826 + 0.563i)5-s + (−0.988 − 0.149i)9-s + (0.988 − 0.149i)11-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)15-s + (−0.955 − 0.294i)17-s + (−0.5 − 0.866i)19-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.222 + 0.974i)27-s + (0.222 + 0.974i)29-s + (0.5 − 0.866i)31-s + (−0.0747 − 0.997i)33-s + (0.733 − 0.680i)37-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)3-s + (0.826 + 0.563i)5-s + (−0.988 − 0.149i)9-s + (0.988 − 0.149i)11-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)15-s + (−0.955 − 0.294i)17-s + (−0.5 − 0.866i)19-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.222 + 0.974i)27-s + (0.222 + 0.974i)29-s + (0.5 − 0.866i)31-s + (−0.0747 − 0.997i)33-s + (0.733 − 0.680i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.725 - 0.687i$
Analytic conductor: \(42.1262\)
Root analytic conductor: \(42.1262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 392,\ (1:\ ),\ 0.725 - 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.388482382 - 0.9521683013i\)
\(L(\frac12)\) \(\approx\) \(2.388482382 - 0.9521683013i\)
\(L(1)\) \(\approx\) \(1.346664005 - 0.3239583541i\)
\(L(1)\) \(\approx\) \(1.346664005 - 0.3239583541i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.0747 - 0.997i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (0.988 - 0.149i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (-0.955 - 0.294i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.733 - 0.680i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (-0.365 + 0.930i)T \)
53 \( 1 + (0.733 + 0.680i)T \)
59 \( 1 + (0.826 - 0.563i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.365 - 0.930i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.988 + 0.149i)T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.67850448791441455504073497041, −23.20560594890115921159465030868, −22.51548060308675934978502400407, −21.5869615400435941533795190583, −20.96181657262140833282791106330, −20.17718124089384803947793189391, −19.35652034796959493296662290676, −17.86708775510866612768943366583, −17.220459452280441379447656418083, −16.48775481594013086533865178932, −15.48619057942126683288716391280, −14.67491117630551114067021091619, −13.714616395161737824091268641247, −12.83597889891689671790263218040, −11.63747580258490869801230544518, −10.62162347131445074190665886387, −9.82426305983796123783453268022, −8.93687336100581120201174745873, −8.274146681435099653366712624295, −6.49468149019207433781847725479, −5.70313425903379653871644560601, −4.63575375473738654062601489031, −3.74286524397953886348721067266, −2.408396406371470694222306061642, −1.00351768375450090258713326181, 0.89674348287052371104533601162, 2.01966670903987352383465030640, 2.92676994493212893185613455095, 4.41627923971650099463081376926, 5.95351088549580293258046597233, 6.578011738536500813989203904499, 7.288627104848450494114175987425, 8.854335839480266807548832952399, 9.23360573108609996399119981590, 10.92139195919997555529573543553, 11.389890567530842356394793299967, 12.694616125770932821592599804384, 13.47993656594871763707852959973, 14.185019730680193631039983651589, 14.962984506313924895275369639938, 16.40932268678703257107630361228, 17.39280751839112579668147837277, 17.92597334060329524795206947821, 18.91453919786050157776051614988, 19.50920014308474896664592969864, 20.60752490867636033340147921165, 21.63074323314846575499473110668, 22.44655785142059423314917022553, 23.259159591212626959743199185774, 24.341088274999583573623959533228

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