↑

Properties

Label 1-291-291.14-r0-0-0
Degree $1$
Conductor $291$
Sign $0.977 - 0.210i$
Analytic cond. $1.35139$
Root an. cond. $1.35139$
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  + (−0.991 − 0.130i)2-s + (0.965 + 0.258i)4-s + (0.442 + 0.896i)5-s + (0.997 − 0.0654i)7-s + (−0.923 − 0.382i)8-s + (−0.321 − 0.946i)10-s + (−0.130 − 0.991i)11-s + (0.896 − 0.442i)13-s + (−0.997 − 0.0654i)14-s + (0.866 + 0.5i)16-s + (0.0654 − 0.997i)17-s + (0.555 − 0.831i)19-s + (0.195 + 0.980i)20-s + i·22-s + (−0.659 − 0.751i)23-s + ⋯
L(s)  = 1  + (−0.991 − 0.130i)2-s + (0.965 + 0.258i)4-s + (0.442 + 0.896i)5-s + (0.997 − 0.0654i)7-s + (−0.923 − 0.382i)8-s + (−0.321 − 0.946i)10-s + (−0.130 − 0.991i)11-s + (0.896 − 0.442i)13-s + (−0.997 − 0.0654i)14-s + (0.866 + 0.5i)16-s + (0.0654 − 0.997i)17-s + (0.555 − 0.831i)19-s + (0.195 + 0.980i)20-s + i·22-s + (−0.659 − 0.751i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(291\)    =    \(3 \cdot 97\)
Sign: $0.977 - 0.210i$
Analytic conductor: \(1.35139\)
Root analytic conductor: \(1.35139\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{291} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 291,\ (0:\ ),\ 0.977 - 0.210i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9944706262 - 0.1058263302i\)
\(L(\frac12)\) \(\approx\) \(0.9944706262 - 0.1058263302i\)
\(L(1)\) \(\approx\) \(0.8705223688 - 0.03149834383i\)
\(L(1)\) \(\approx\) \(0.8705223688 - 0.03149834383i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
97 \( 1 \)
good2 \( 1 + (-0.991 - 0.130i)T \)
5 \( 1 + (0.442 + 0.896i)T \)
7 \( 1 + (0.997 - 0.0654i)T \)
11 \( 1 + (-0.130 - 0.991i)T \)
13 \( 1 + (0.896 - 0.442i)T \)
17 \( 1 + (0.0654 - 0.997i)T \)
19 \( 1 + (0.555 - 0.831i)T \)
23 \( 1 + (-0.659 - 0.751i)T \)
29 \( 1 + (-0.321 + 0.946i)T \)
31 \( 1 + (0.608 + 0.793i)T \)
37 \( 1 + (-0.751 - 0.659i)T \)
41 \( 1 + (0.946 + 0.321i)T \)
43 \( 1 + (-0.258 - 0.965i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (-0.130 + 0.991i)T \)
59 \( 1 + (0.659 - 0.751i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.831 - 0.555i)T \)
71 \( 1 + (0.946 - 0.321i)T \)
73 \( 1 + (0.965 - 0.258i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (0.997 + 0.0654i)T \)
89 \( 1 + (0.923 + 0.382i)T \)
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

−25.644610556134610999560184504268, −24.62354717532135558919912089406, −24.10942595635915094675131449089, −23.08949962802631500459750892467, −21.38965665342441314186966061005, −20.85097787238538255952071182018, −20.18359704312396892572866622031, −19.04729779269714206087597791480, −17.95893037831026926045613024097, −17.46580584052065404415331242701, −16.56501727007158451352977079514, −15.59887915347565099202958765536, −14.66414990526429256985845823882, −13.48032204760118111594573633381, −12.19942568554906341533288030035, −11.45200430274239835928744002029, −10.20493937646230353500558542027, −9.45106753553184973226504458661, −8.314609795634838641713012897546, −7.80398446334763893475216844357, −6.29361221845573663170032751046, −5.36580054679833442317315384191, −4.04347032727905246412108146155, −1.99394627002861923957736676931, −1.40351014475417740525818811836, 1.08942711000330268963456137140, 2.46124084098469085853018618768, 3.416389268751862860492812712666, 5.33335318950722581811118188740, 6.42934155229403887216202193497, 7.442595856379551864695586788280, 8.399136713478787027312804749407, 9.33340060117907590416101905558, 10.72187099285069319784828995033, 10.9364729963228645894536963901, 12.01466109534054355501094082681, 13.60984488077877833269181888598, 14.3786363095073794868108708338, 15.56451925422903945509652886059, 16.348290660717920103527004136587, 17.64607491528466378783360901410, 18.12142079458068823329917327024, 18.785796100867322282754139175867, 19.96906436977231873001563798561, 20.89668754068024645101468886714, 21.57041784157991315594903942036, 22.62891513399061431080443503222, 23.956715809452233227795641169278, 24.7063803812626035699417965528, 25.63255164719441728239024083937

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