↑

Properties

Label 1-320-320.19-r1-0-0
Degree $1$
Conductor $320$
Sign $0.956 + 0.290i$
Analytic cond. $34.3887$
Root an. cond. $34.3887$
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  + (−0.382 + 0.923i)3-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.923 + 0.382i)13-s − i·17-s + (−0.923 − 0.382i)19-s + (−0.382 − 0.923i)21-s + (0.707 + 0.707i)23-s + (0.923 − 0.382i)27-s + (0.382 − 0.923i)29-s + 31-s + 33-s + (−0.923 + 0.382i)37-s + (−0.707 + 0.707i)39-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.923 + 0.382i)13-s − i·17-s + (−0.923 − 0.382i)19-s + (−0.382 − 0.923i)21-s + (0.707 + 0.707i)23-s + (0.923 − 0.382i)27-s + (0.382 − 0.923i)29-s + 31-s + 33-s + (−0.923 + 0.382i)37-s + (−0.707 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(34.3887\)
Root analytic conductor: \(34.3887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (1:\ ),\ 0.956 + 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.261670526 + 0.1871511434i\)
\(L(\frac12)\) \(\approx\) \(1.261670526 + 0.1871511434i\)
\(L(1)\) \(\approx\) \(0.8514718062 + 0.1973943636i\)
\(L(1)\) \(\approx\) \(0.8514718062 + 0.1973943636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.923 - 0.382i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (0.923 - 0.382i)T \)
61 \( 1 + (-0.382 + 0.923i)T \)
67 \( 1 + (-0.382 + 0.923i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.923 + 0.382i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
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

−24.939080300639794955757129448, −23.76962806270050619228253054554, −23.10575462375228174147458276873, −22.67128385341651300930169041523, −21.256872813568953338944269651, −20.24582008260074668378364508171, −19.41896771380671237001053430395, −18.62081962891460764148917614120, −17.624147607191316857534652672377, −16.95448637986183637464201125063, −15.95152446531141349203141686193, −14.79376438079456658366736027200, −13.65044052799470550770867877367, −12.83106959429180888517880928209, −12.34207393592752334328415606982, −10.79769675856639264016232195873, −10.36135563144984085200488479414, −8.76852965482827963555625774170, −7.806120398648787033022906917293, −6.76239285644746841305935634279, −6.11725116526159221473187507193, −4.74331095170031757725408466841, −3.41726200913390144996192096359, −2.03038638484549658583572609733, −0.79188396398584276323553182790, 0.577251293925258269698713123459, 2.67604689427898380505967368879, 3.578777717549072549190300146287, 4.84180174560481189292565267440, 5.85685903806510216341184670650, 6.625958840119099123871392355812, 8.40418396388807739314914456380, 9.11696416703605172825516747499, 10.06372309433691505796078946204, 11.1494182864935971679203668147, 11.78095514721905109106555538622, 13.09646409704100710451175930097, 14.00024602293154296480833777988, 15.364451085846360060746921068231, 15.8172246741997988896234854099, 16.62157211287612170638672130880, 17.66069332811500379084556495665, 18.76523456965256727265855583506, 19.476070482184551783453299370130, 20.98417948835500821174547096062, 21.21964613672439080317354487735, 22.31491358999735967286039632609, 23.00106409660497159535123464670, 23.90312782360282101776161187860, 25.16701085106717287423623541103

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