↑

Properties

Label 1-1617-1617.515-r1-0-0
Degree $1$
Conductor $1617$
Sign $0.720 + 0.693i$
Analytic cond. $173.770$
Root an. cond. $173.770$
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  + (0.999 + 0.0299i)2-s + (0.998 + 0.0598i)4-s + (0.946 − 0.323i)5-s + (0.995 + 0.0896i)8-s + (0.955 − 0.294i)10-s + (0.473 + 0.880i)13-s + (0.992 + 0.119i)16-s + (−0.887 + 0.460i)17-s + (0.669 + 0.743i)19-s + (0.963 − 0.266i)20-s + (0.988 − 0.149i)23-s + (0.791 − 0.611i)25-s + (0.447 + 0.894i)26-s + (−0.936 − 0.351i)29-s + (−0.104 + 0.994i)31-s + (0.988 + 0.149i)32-s + ⋯
L(s)  = 1  + (0.999 + 0.0299i)2-s + (0.998 + 0.0598i)4-s + (0.946 − 0.323i)5-s + (0.995 + 0.0896i)8-s + (0.955 − 0.294i)10-s + (0.473 + 0.880i)13-s + (0.992 + 0.119i)16-s + (−0.887 + 0.460i)17-s + (0.669 + 0.743i)19-s + (0.963 − 0.266i)20-s + (0.988 − 0.149i)23-s + (0.791 − 0.611i)25-s + (0.447 + 0.894i)26-s + (−0.936 − 0.351i)29-s + (−0.104 + 0.994i)31-s + (0.988 + 0.149i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.720 + 0.693i$
Analytic conductor: \(173.770\)
Root analytic conductor: \(173.770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (1:\ ),\ 0.720 + 0.693i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.821534059 + 2.345385546i\)
\(L(\frac12)\) \(\approx\) \(5.821534059 + 2.345385546i\)
\(L(1)\) \(\approx\) \(2.532541415 + 0.3071344931i\)
\(L(1)\) \(\approx\) \(2.532541415 + 0.3071344931i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.999 + 0.0299i)T \)
5 \( 1 + (0.946 - 0.323i)T \)
13 \( 1 + (0.473 + 0.880i)T \)
17 \( 1 + (-0.887 + 0.460i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (-0.936 - 0.351i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.772 + 0.635i)T \)
41 \( 1 + (0.995 + 0.0896i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.280 + 0.959i)T \)
53 \( 1 + (0.842 - 0.538i)T \)
59 \( 1 + (-0.575 - 0.817i)T \)
61 \( 1 + (-0.842 - 0.538i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.963 + 0.266i)T \)
73 \( 1 + (-0.280 + 0.959i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (-0.473 + 0.880i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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

−20.39576450013169829323634059793, −19.70634733658065300109570424755, −18.639398787122741132644680436249, −17.91323019417400244638159722256, −17.1458288267183099877380382866, −16.35182439701924110715653727540, −15.33620393685525862357896127388, −15.04026309208207431042021668342, −13.933237624145865053746176706947, −13.42647096248225787642744486155, −12.95510660991047534591299003692, −11.93331794058814729280023118014, −10.96063791262581324162985271695, −10.652118203133956312153458004384, −9.53178512167473402145443335723, −8.77583053521647034581090949771, −7.36445063786422921155699735265, −6.9850035915026529457010642635, −5.83210351715432723828664605784, −5.47790466405448897416473004408, −4.51264675277131711781061164137, −3.41915236290557655262808131340, −2.69546931160012205281226330061, −1.88834296582617356312270970992, −0.74427479066133175141211271468, 1.24638524108420794449348695179, 1.873000768177882783786112441198, 2.86659889687586293757729261443, 3.84643996187220754172306295012, 4.7229280505777832209449478343, 5.44241403627998765780295007704, 6.307658931404128111894957041052, 6.80788814501564330219644103152, 7.92585298086495955359294390680, 8.93622600755049646638513151819, 9.6848483369962831587334757589, 10.72743379952272585967525424476, 11.261552757567120057620794598355, 12.30589224223524062207215035067, 12.90120145335006481016702017646, 13.666823947160121089720203148482, 14.15283974786004185873195581700, 14.9622250794474439463379883817, 15.84826388514993842500768213580, 16.564151437198012406325564657249, 17.15257815003145842314512865499, 18.07075040787400108674080638309, 18.97099466174461930135734738806, 19.83061121770926222947021023302, 20.67510709201198883518394797232

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