↑

Properties

Label 1-1337-1337.40-r1-0-0
Degree $1$
Conductor $1337$
Sign $-0.978 - 0.204i$
Analytic cond. $143.680$
Root an. cond. $143.680$
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  + (0.391 − 0.920i)2-s + (0.949 − 0.314i)3-s + (−0.693 − 0.720i)4-s + (0.926 + 0.376i)5-s + (0.0825 − 0.996i)6-s + (−0.934 + 0.355i)8-s + (0.802 − 0.596i)9-s + (0.709 − 0.705i)10-s + (0.0275 − 0.999i)11-s + (−0.884 − 0.466i)12-s + (0.213 − 0.976i)13-s + (0.997 + 0.0660i)15-s + (−0.0385 + 0.999i)16-s + (−0.266 − 0.963i)17-s + (−0.234 − 0.972i)18-s + (0.709 + 0.705i)19-s + ⋯
L(s)  = 1  + (0.391 − 0.920i)2-s + (0.949 − 0.314i)3-s + (−0.693 − 0.720i)4-s + (0.926 + 0.376i)5-s + (0.0825 − 0.996i)6-s + (−0.934 + 0.355i)8-s + (0.802 − 0.596i)9-s + (0.709 − 0.705i)10-s + (0.0275 − 0.999i)11-s + (−0.884 − 0.466i)12-s + (0.213 − 0.976i)13-s + (0.997 + 0.0660i)15-s + (−0.0385 + 0.999i)16-s + (−0.266 − 0.963i)17-s + (−0.234 − 0.972i)18-s + (0.709 + 0.705i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1337\)    =    \(7 \cdot 191\)
Sign: $-0.978 - 0.204i$
Analytic conductor: \(143.680\)
Root analytic conductor: \(143.680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1337} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1337,\ (1:\ ),\ -0.978 - 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4540211760 - 4.390371065i\)
\(L(\frac12)\) \(\approx\) \(0.4540211760 - 4.390371065i\)
\(L(1)\) \(\approx\) \(1.398707683 - 1.422180752i\)
\(L(1)\) \(\approx\) \(1.398707683 - 1.422180752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.391 - 0.920i)T \)
3 \( 1 + (0.949 - 0.314i)T \)
5 \( 1 + (0.926 + 0.376i)T \)
11 \( 1 + (0.0275 - 0.999i)T \)
13 \( 1 + (0.213 - 0.976i)T \)
17 \( 1 + (-0.266 - 0.963i)T \)
19 \( 1 + (0.709 + 0.705i)T \)
23 \( 1 + (0.988 + 0.153i)T \)
29 \( 1 + (-0.768 - 0.639i)T \)
31 \( 1 + (-0.975 + 0.218i)T \)
37 \( 1 + (0.975 + 0.218i)T \)
41 \( 1 + (-0.945 - 0.324i)T \)
43 \( 1 + (-0.518 - 0.854i)T \)
47 \( 1 + (0.997 - 0.0770i)T \)
53 \( 1 + (0.959 - 0.282i)T \)
59 \( 1 + (-0.509 - 0.860i)T \)
61 \( 1 + (-0.999 + 0.0440i)T \)
67 \( 1 + (-0.0605 + 0.998i)T \)
71 \( 1 + (-0.956 + 0.293i)T \)
73 \( 1 + (0.381 - 0.924i)T \)
79 \( 1 + (0.938 + 0.345i)T \)
83 \( 1 + (0.627 + 0.778i)T \)
89 \( 1 + (-0.329 + 0.944i)T \)
97 \( 1 + (-0.980 + 0.197i)T \)
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

−21.28753406886859843102232403818, −20.434994177559650705071737250148, −19.76422558265826232225390227416, −18.50905339925234779366041912653, −18.07120001568610074544350312103, −16.91032928344551117518134519182, −16.630293302936280000118883872724, −15.54464940962129253790303316180, −14.87177917757134625051746355066, −14.37012009335750429616562550407, −13.36064331022561453598390522457, −13.13810718268098465252017766013, −12.200155914019048858658543354727, −10.84291720973090770315441304164, −9.733087023415058676682595100341, −9.15441756965376412692923720635, −8.71733382279107248740825765790, −7.53401640853539934842270664632, −6.94554131747834294862970003420, −5.97949882622394851935259017717, −4.91285673549508436892267986499, −4.42054033784354765950067244754, −3.39929846384004247881833164432, −2.32971867413483553362851154522, −1.40197946686795129061171980843, 0.615299894269702548330513972399, 1.481620722021844624181641794717, 2.47534968149657544966395372475, 3.13038865579849925026847593595, 3.75446875523713572972292936163, 5.195431497110754391953870194841, 5.78323154903287337736241030825, 6.84290880912410386574566406522, 7.89392964243944395872265873013, 8.92366380884541639554443447153, 9.41902850291275179344258232404, 10.26954151370573540122091008726, 10.980482036933492513023368856907, 11.93118163460338473597939324814, 12.9340414561211702891353476033, 13.538148532288081282041865503142, 13.87461155320716835341366901200, 14.78606125406532872726994625736, 15.37498400237664990088983824802, 16.62799780818310430685006996467, 17.73802972680209085094723153218, 18.55453003441595653557137216789, 18.67663054141619711420089418078, 19.77331557767219042513768076808, 20.56232886090701341977972763710

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