↑

Properties

Label 1-392-392.5-r1-0-0
Degree $1$
Conductor $392$
Sign $-0.695 - 0.718i$
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  + (0.365 + 0.930i)3-s + (−0.988 − 0.149i)5-s + (−0.733 + 0.680i)9-s + (0.733 + 0.680i)11-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)27-s + (0.900 − 0.433i)29-s + (0.5 − 0.866i)31-s + (−0.365 + 0.930i)33-s + (−0.826 − 0.563i)37-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)3-s + (−0.988 − 0.149i)5-s + (−0.733 + 0.680i)9-s + (0.733 + 0.680i)11-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)27-s + (0.900 − 0.433i)29-s + (0.5 − 0.866i)31-s + (−0.365 + 0.930i)33-s + (−0.826 − 0.563i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.695 - 0.718i$
Analytic conductor: \(42.1262\)
Root analytic conductor: \(42.1262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 392,\ (1:\ ),\ -0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1827446655 + 0.4313735935i\)
\(L(\frac12)\) \(\approx\) \(-0.1827446655 + 0.4313735935i\)
\(L(1)\) \(\approx\) \(0.7544452710 + 0.3818649094i\)
\(L(1)\) \(\approx\) \(0.7544452710 + 0.3818649094i\)

Euler product

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

−23.66540778926706980302272027845, −22.95814853287500026329006274575, −22.24163261718848855469568252286, −20.76737985220688597007857409755, −20.03691462779591019619006406166, −19.275516867027320971391910267092, −18.63193409399121338540510374619, −17.71103926419546971337722379007, −16.6416470387496379369062325636, −15.67651886794458264401481619062, −14.60681532354011870443415633468, −14.02963261745952742099171379841, −12.7871841423164131999557024362, −12.1071553275323161698000341170, −11.31844078125154960764488664263, −10.14756485841558724933321898367, −8.598356907429478927425667038508, −8.245588602032002837778083922937, −7.07595977182657052135248455285, −6.3781576181833420169715528508, −4.957506594444196135357586449278, −3.534575480544481868294997873704, −2.81323868605646931536406369070, −1.20707043560455222582158515104, −0.13057122950604092307072011694, 1.81219043941298142945219458802, 3.26034800204770847057504308138, 4.2375447367217243911095526118, 4.75530157837189662399993657262, 6.358490496831730620580038181377, 7.48523452372599068713161830703, 8.542941457200768371676944621352, 9.26677273757745167709240632800, 10.2804800946394750076725134335, 11.37303214765820052963228682511, 11.979870912736746973785204993255, 13.27274164465785700495922059825, 14.41364316898091192176187207013, 15.1620407806076599690953681058, 15.75455403592319781534101865289, 16.82504586148327408643949165318, 17.43992392833067329457352652162, 19.154883711133879297023743547177, 19.525388833335894953483936255348, 20.345959600979960236215836638646, 21.37859333600763923900936970662, 22.03148610062437754737075145883, 23.0483086322322581290617026450, 23.77831672360978073884044070349, 24.81867748792799646936413099485

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