Properties

Label 1-297-297.191-r1-0-0
Degree $1$
Conductor $297$
Sign $0.0354 - 0.999i$
Analytic cond. $31.9170$
Root an. cond. $31.9170$
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  + (−0.0348 + 0.999i)2-s + (−0.997 − 0.0697i)4-s + (0.882 + 0.469i)5-s + (−0.241 + 0.970i)7-s + (0.104 − 0.994i)8-s + (−0.5 + 0.866i)10-s + (−0.615 − 0.788i)13-s + (−0.961 − 0.275i)14-s + (0.990 + 0.139i)16-s + (−0.669 − 0.743i)17-s + (−0.104 + 0.994i)19-s + (−0.848 − 0.529i)20-s + (−0.766 − 0.642i)23-s + (0.559 + 0.829i)25-s + (0.809 − 0.587i)26-s + ⋯
L(s)  = 1  + (−0.0348 + 0.999i)2-s + (−0.997 − 0.0697i)4-s + (0.882 + 0.469i)5-s + (−0.241 + 0.970i)7-s + (0.104 − 0.994i)8-s + (−0.5 + 0.866i)10-s + (−0.615 − 0.788i)13-s + (−0.961 − 0.275i)14-s + (0.990 + 0.139i)16-s + (−0.669 − 0.743i)17-s + (−0.104 + 0.994i)19-s + (−0.848 − 0.529i)20-s + (−0.766 − 0.642i)23-s + (0.559 + 0.829i)25-s + (0.809 − 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0354 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0354 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.0354 - 0.999i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ 0.0354 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06203114728 + 0.05987264858i\)
\(L(\frac12)\) \(\approx\) \(-0.06203114728 + 0.05987264858i\)
\(L(1)\) \(\approx\) \(0.6594487750 + 0.4709018815i\)
\(L(1)\) \(\approx\) \(0.6594487750 + 0.4709018815i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0348 + 0.999i)T \)
5 \( 1 + (0.882 + 0.469i)T \)
7 \( 1 + (-0.241 + 0.970i)T \)
13 \( 1 + (-0.615 - 0.788i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.961 + 0.275i)T \)
31 \( 1 + (-0.374 + 0.927i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (-0.961 - 0.275i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.997 - 0.0697i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.438 - 0.898i)T \)
61 \( 1 + (-0.374 - 0.927i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (0.615 - 0.788i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.882 + 0.469i)T \)
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

−24.12207892547836746808120017137, −23.74658143827441262857525756724, −22.22833374703099743853663625781, −21.87620082529831277882520018791, −20.77510332782445567916984545224, −20.08096389427845648771334798084, −19.31535399475816502279111074708, −18.17939197923549925175542942975, −17.13652572428679444928622770679, −16.85028295394673769914545148076, −15.119856488481949061669819244625, −13.75214208337530102761336655698, −13.499630133611480279727855458628, −12.451809433244394869218646642834, −11.34149944075284273703508824024, −10.35431988842953516804908942579, −9.571780338558292155315973003820, −8.77278232687029091738590438342, −7.36973685996838979772162571018, −6.01760134550666228107158243087, −4.73080048475571027533941319600, −3.92071895001589205153806097497, −2.4062107360932402547670262018, −1.43498606871671120651436528696, −0.02521525111280416878522354463, 2.02671817910122945067677240263, 3.33147481384576063725853021170, 4.9913936368309278742962762529, 5.76939745354063986438345242208, 6.60455852059500043925046252234, 7.71689920351606231385215872204, 8.89054316945398657938035118364, 9.65754734752391158223745654460, 10.605657997651577965602602293933, 12.263952289960688718094742590427, 13.05978173946476631555621353635, 14.187204902405274073166038134114, 14.81445669870170866859514902716, 15.77868328355813567478915315875, 16.685539220642761034949549250450, 17.82057017749001475595361154581, 18.24435558269798167111836067660, 19.17213841770976917200538856250, 20.562793319307544661004953202067, 21.862164452966104420411113370504, 22.24604742383560766949641296088, 23.08422181426831189776541573470, 24.499420126977219286204227731731, 24.94038265308695025614392883181, 25.65378482800388112350059355697

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