↑

Properties

Label 1-920-920.219-r1-0-0
Degree $1$
Conductor $920$
Sign $0.684 + 0.728i$
Analytic cond. $98.8677$
Root an. cond. $98.8677$
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  + (0.959 + 0.281i)3-s + (−0.142 + 0.989i)7-s + (0.841 + 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.142 − 0.989i)13-s + (0.654 − 0.755i)17-s + (−0.654 − 0.755i)19-s + (−0.415 + 0.909i)21-s + (0.654 + 0.755i)27-s + (0.654 − 0.755i)29-s + (0.959 − 0.281i)31-s + (0.142 + 0.989i)33-s + (0.841 + 0.540i)37-s + (0.142 − 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)3-s + (−0.142 + 0.989i)7-s + (0.841 + 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.142 − 0.989i)13-s + (0.654 − 0.755i)17-s + (−0.654 − 0.755i)19-s + (−0.415 + 0.909i)21-s + (0.654 + 0.755i)27-s + (0.654 − 0.755i)29-s + (0.959 − 0.281i)31-s + (0.142 + 0.989i)33-s + (0.841 + 0.540i)37-s + (0.142 − 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.684 + 0.728i$
Analytic conductor: \(98.8677\)
Root analytic conductor: \(98.8677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (1:\ ),\ 0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.231645149 + 1.397498033i\)
\(L(\frac12)\) \(\approx\) \(3.231645149 + 1.397498033i\)
\(L(1)\) \(\approx\) \(1.609369812 + 0.3456477197i\)
\(L(1)\) \(\approx\) \(1.609369812 + 0.3456477197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (-0.142 + 0.989i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (0.654 - 0.755i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (0.841 + 0.540i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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

−21.150275986566391859935934498108, −21.051922102886460543253589357048, −19.67204656366280814747947426770, −19.4504467020194184942645183304, −18.702509749309045944571203081122, −17.64495157288272058089569233735, −16.66128074781571806645642690443, −16.203129760864094579141097712956, −14.93280297022032285578497815002, −14.20551851825830281195301864601, −13.82772868394626020163130720149, −12.837382408199837312855277231737, −12.08591689546538223821519156050, −10.90711662565853218932862839278, −10.14059300164944135056961581644, −9.229589673896430280033423212277, −8.41267581437073406738670163223, −7.666347571221183147387413111881, −6.744095264266978269834040536822, −6.0188071727518172714674686904, −4.371372915881205172122647682885, −3.83435997375195275103241068799, −2.88695940778075403476311057803, −1.654233234037086981989102281278, −0.8166564170339772275081853790, 0.96625934141870151212203823291, 2.48212067717789164738064281818, 2.71624421589408166384682746438, 4.08388983866260037055596186000, 4.88572727051152111905737112322, 5.93871376799012613294413141974, 7.07850841446651222369432142419, 7.930321088471486524557447778034, 8.717287509826089119531020646576, 9.5853072616524841894724164820, 10.06422193742855132685375286431, 11.28621794091560102145874802861, 12.34788569272762590782289351314, 12.87069850637809915777190290101, 13.92558883991890120758475218953, 14.69797900627269029856095417026, 15.46736439049571521285806091101, 15.78362506073832792699718588893, 17.12085933289525649226138599499, 17.893524313047088707095883118264, 18.83556455862471949780096996290, 19.41996441755284714803424959412, 20.28558797984635646396573551932, 20.88206446458611133871109632468, 21.74379880402954711012501481573

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