Properties

Label 1-33e2-1089.619-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.941 + 0.336i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (−0.0665 − 0.997i)2-s + (−0.991 + 0.132i)4-s + (−0.217 − 0.976i)5-s + (0.449 + 0.893i)7-s + (0.198 + 0.980i)8-s + (−0.959 + 0.281i)10-s + (−0.432 − 0.901i)13-s + (0.861 − 0.508i)14-s + (0.964 − 0.263i)16-s + (−0.921 − 0.389i)17-s + (0.974 + 0.226i)19-s + (0.345 + 0.938i)20-s + (−0.888 + 0.458i)23-s + (−0.905 + 0.424i)25-s + (−0.870 + 0.491i)26-s + ⋯
L(s)  = 1  + (−0.0665 − 0.997i)2-s + (−0.991 + 0.132i)4-s + (−0.217 − 0.976i)5-s + (0.449 + 0.893i)7-s + (0.198 + 0.980i)8-s + (−0.959 + 0.281i)10-s + (−0.432 − 0.901i)13-s + (0.861 − 0.508i)14-s + (0.964 − 0.263i)16-s + (−0.921 − 0.389i)17-s + (0.974 + 0.226i)19-s + (0.345 + 0.938i)20-s + (−0.888 + 0.458i)23-s + (−0.905 + 0.424i)25-s + (−0.870 + 0.491i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.941 + 0.336i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.941 + 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1157921418 - 0.6679521402i\)
\(L(\frac12)\) \(\approx\) \(-0.1157921418 - 0.6679521402i\)
\(L(1)\) \(\approx\) \(0.6092190169 - 0.5026592911i\)
\(L(1)\) \(\approx\) \(0.6092190169 - 0.5026592911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0665 - 0.997i)T \)
5 \( 1 + (-0.217 - 0.976i)T \)
7 \( 1 + (0.449 + 0.893i)T \)
13 \( 1 + (-0.432 - 0.901i)T \)
17 \( 1 + (-0.921 - 0.389i)T \)
19 \( 1 + (0.974 + 0.226i)T \)
23 \( 1 + (-0.888 + 0.458i)T \)
29 \( 1 + (0.820 - 0.572i)T \)
31 \( 1 + (-0.851 - 0.524i)T \)
37 \( 1 + (0.941 + 0.336i)T \)
41 \( 1 + (0.999 + 0.0380i)T \)
43 \( 1 + (-0.995 + 0.0950i)T \)
47 \( 1 + (0.272 - 0.962i)T \)
53 \( 1 + (-0.254 - 0.967i)T \)
59 \( 1 + (-0.532 - 0.846i)T \)
61 \( 1 + (-0.830 - 0.556i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (-0.362 - 0.931i)T \)
79 \( 1 + (-0.595 - 0.803i)T \)
83 \( 1 + (0.161 - 0.986i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (-0.217 + 0.976i)T \)
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

−22.02200226705678345257601956506, −21.44081342113696112662132308182, −19.98197779305025017427697170742, −19.576800075863337813969267232659, −18.37447955657088263584835085222, −18.02668463877272169224270599579, −17.19359551881558473888296530193, −16.34663676336727878562093857014, −15.686831342216338684711945895015, −14.705875244870734416890747038148, −14.18053067935334511131869238228, −13.68139697862355618827238378089, −12.54453230045188974955861066905, −11.43551574411775828421715623011, −10.64776327281881750062037018189, −9.863021563003172408646053997787, −8.93071757879929414086348317545, −7.91663878405568871248603561509, −7.21457372919169339573812743605, −6.70151209508804389631277056486, −5.75114181656058883910396287308, −4.47922954041654431332666856684, −4.06901665146811942760471213622, −2.79448671139151277739918416044, −1.36978442870335761953408558700, 0.30216959109644923033482105251, 1.55809906125645527525252393207, 2.413164767893317796106706988571, 3.44154268718955449157924354216, 4.55758115148015110494555560042, 5.15049749461573506203447746884, 5.96184561639460169035569769715, 7.781007769391575784870209628140, 8.166560488051029672329495269129, 9.23113161174325249713437440279, 9.63155883387981270599895525859, 10.78705129706397158631768228275, 11.79486051571976333910575555647, 12.03063674864140512403792436324, 13.01266131777758531434103740441, 13.602746200244777881482144208750, 14.68547543273076231565188194491, 15.535509603219673323070388813025, 16.36772364263434752864657284360, 17.46602807226657365450214123912, 17.957788598427023888767551833510, 18.68426962181743670498555983579, 19.904013006111736163715776479635, 20.021187405233177657663993695881, 20.917131986376084009904296797799

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