↑

Properties

Label 1-684-684.499-r1-0-0
Degree $1$
Conductor $684$
Sign $0.977 + 0.211i$
Analytic cond. $73.5060$
Root an. cond. $73.5060$
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  + (0.766 − 0.642i)5-s + (0.5 + 0.866i)7-s − 11-s + (0.766 + 0.642i)13-s + (0.766 − 0.642i)17-s + (0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.939 + 0.342i)29-s − 31-s + (0.939 + 0.342i)35-s + 37-s + (0.173 + 0.984i)41-s + (0.939 + 0.342i)43-s + (0.939 − 0.342i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)5-s + (0.5 + 0.866i)7-s − 11-s + (0.766 + 0.642i)13-s + (0.766 − 0.642i)17-s + (0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.939 + 0.342i)29-s − 31-s + (0.939 + 0.342i)35-s + 37-s + (0.173 + 0.984i)41-s + (0.939 + 0.342i)43-s + (0.939 − 0.342i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(73.5060\)
Root analytic conductor: \(73.5060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (1:\ ),\ 0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.696957272 + 0.2890068872i\)
\(L(\frac12)\) \(\approx\) \(2.696957272 + 0.2890068872i\)
\(L(1)\) \(\approx\) \(1.372418631 + 0.02626428899i\)
\(L(1)\) \(\approx\) \(1.372418631 + 0.02626428899i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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

−22.54089790896023046086098527766, −21.54456275850016951405654747679, −20.85032216156231341880788324012, −20.316270723351087558001082334412, −18.98182161729492447002513223119, −18.42126258406075761606291622965, −17.51105190669022852441814564781, −16.983910858357331494431244473784, −15.847387318815141512995528895920, −14.91726991391896284359015170035, −14.214134946225374632393422000558, −13.29241243098993601534342932676, −12.814140202397716098026619203657, −11.22529602009977145478244123356, −10.702480525247836489942528422085, −10.066148350980142236834301063464, −8.94522024175420834384546580420, −7.75348513332837225052411993806, −7.2463275847935162738603738514, −5.9077671803595562447804764700, −5.371372861553292581389485296175, −3.986399894110043496755875830244, −3.05788462417849015061358775604, −1.912077200380169942972197369675, −0.79202971229917346576517876767, 0.929634238301386045176990777320, 1.999604831840384096230921610763, 2.90254408698150526975546846843, 4.38845766168410072296321347380, 5.387414380021926915891790095350, 5.80377669736800569054133792438, 7.150662584632048944785208370140, 8.23572065361762376132730494116, 8.99132051261177613438189054510, 9.69345242723100244145845196966, 10.84777643223231762336254064967, 11.63444640769835935791790552234, 12.720845818303262817582823923797, 13.23775225008242652339892019822, 14.28344613845766280799320464752, 15.02877799408269001217985568367, 16.198480658581393240422469004438, 16.579461449642208565585972084696, 17.85456457194989120192819393199, 18.327646132978604781096679438109, 19.06279729724673887896796619074, 20.47802993383662604033124685772, 20.93157187924088692789640662190, 21.50269962103007543306622537874, 22.4223757744366664891240659977

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