↑

Properties

Label 1-297-297.200-r0-0-0
Degree $1$
Conductor $297$
Sign $0.299 - 0.954i$
Analytic cond. $1.37926$
Root an. cond. $1.37926$
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  + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.997 + 0.0697i)5-s + (−0.0348 + 0.999i)7-s + (−0.978 + 0.207i)8-s + (0.5 − 0.866i)10-s + (0.719 − 0.694i)13-s + (0.882 + 0.469i)14-s + (−0.241 + 0.970i)16-s + (−0.104 − 0.994i)17-s + (0.978 − 0.207i)19-s + (−0.559 − 0.829i)20-s + (0.939 − 0.342i)23-s + (0.990 + 0.139i)25-s + (−0.309 − 0.951i)26-s + ⋯
L(s)  = 1  + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.997 + 0.0697i)5-s + (−0.0348 + 0.999i)7-s + (−0.978 + 0.207i)8-s + (0.5 − 0.866i)10-s + (0.719 − 0.694i)13-s + (0.882 + 0.469i)14-s + (−0.241 + 0.970i)16-s + (−0.104 − 0.994i)17-s + (0.978 − 0.207i)19-s + (−0.559 − 0.829i)20-s + (0.939 − 0.342i)23-s + (0.990 + 0.139i)25-s + (−0.309 − 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.299 - 0.954i$
Analytic conductor: \(1.37926\)
Root analytic conductor: \(1.37926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (0:\ ),\ 0.299 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.404527172 - 1.031596089i\)
\(L(\frac12)\) \(\approx\) \(1.404527172 - 1.031596089i\)
\(L(1)\) \(\approx\) \(1.292779712 - 0.6443199199i\)
\(L(1)\) \(\approx\) \(1.292779712 - 0.6443199199i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.438 - 0.898i)T \)
5 \( 1 + (0.997 + 0.0697i)T \)
7 \( 1 + (-0.0348 + 0.999i)T \)
13 \( 1 + (0.719 - 0.694i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.882 + 0.469i)T \)
31 \( 1 + (0.961 + 0.275i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (-0.882 - 0.469i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.615 - 0.788i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.374 - 0.927i)T \)
61 \( 1 + (-0.961 + 0.275i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (-0.438 + 0.898i)T \)
83 \( 1 + (-0.719 - 0.694i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.997 + 0.0697i)T \)
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

−25.638232048285094470202873117617, −24.6097100891218613750215356775, −23.89079109649722090696822277543, −22.995046188338837900714076840819, −22.1584220960854496645884268504, −21.15367443421065987502799495674, −20.58088489823186163419902704749, −19.05351547556389238656507420759, −18.01892928714355420993773417313, −17.08934513654327840314593731497, −16.68022316255002883048719711391, −15.48778689941938777365271501858, −14.42937215311432251862820112088, −13.53715449380780178434025547108, −13.221025371408769446500018799421, −11.79084613461161507743047806408, −10.46723916029688932507132805876, −9.44390856308633985164940805859, −8.45662798265228823780723932119, −7.239679410050579366192473851955, −6.411197119588251285109180317724, −5.46285395288498208666982786550, −4.31463193527452718863632950722, −3.26094274599546926700751653736, −1.45614762242950874673891094400, 1.26134056654454969141545415057, 2.51526720876764470313530530127, 3.28593872489525045527563746834, 5.0796732943432136998538634251, 5.54403129187255697737579686477, 6.74552266195831832969460972301, 8.62966329182713257554747216320, 9.33919071620117747342542858493, 10.27947179420189165000449253591, 11.28466514811109544324604993023, 12.23662699943480037887264845209, 13.21352662444693770301163704599, 13.870188022139380651270202112441, 14.9422903744145244951133498777, 15.842702032987540869749524176657, 17.34551058761012307257895078890, 18.360463317018568972814650361582, 18.63563246160129511939117219823, 20.10182150966512939617901181079, 20.8120017860996335130707046615, 21.588487217351474255464005801338, 22.41667889412787230075649680363, 23.00989551277378193834082729010, 24.519414820613054147537727379514, 24.95471072347500763950349175852

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