↑

Properties

Label 1-1441-1441.25-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.439 + 0.898i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (−0.998 − 0.0483i)2-s + (−0.607 + 0.794i)3-s + (0.995 + 0.0965i)4-s + (0.0241 − 0.999i)5-s + (0.644 − 0.764i)6-s + (−0.970 − 0.239i)7-s + (−0.989 − 0.144i)8-s + (−0.262 − 0.964i)9-s + (−0.0724 + 0.997i)10-s + (−0.681 + 0.732i)12-s + (−0.970 − 0.239i)13-s + (0.958 + 0.285i)14-s + (0.779 + 0.626i)15-s + (0.981 + 0.192i)16-s + (−0.681 − 0.732i)17-s + (0.215 + 0.976i)18-s + ⋯
L(s)  = 1  + (−0.998 − 0.0483i)2-s + (−0.607 + 0.794i)3-s + (0.995 + 0.0965i)4-s + (0.0241 − 0.999i)5-s + (0.644 − 0.764i)6-s + (−0.970 − 0.239i)7-s + (−0.989 − 0.144i)8-s + (−0.262 − 0.964i)9-s + (−0.0724 + 0.997i)10-s + (−0.681 + 0.732i)12-s + (−0.970 − 0.239i)13-s + (0.958 + 0.285i)14-s + (0.779 + 0.626i)15-s + (0.981 + 0.192i)16-s + (−0.681 − 0.732i)17-s + (0.215 + 0.976i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.439 + 0.898i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.439 + 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2224121021 + 0.1387832875i\)
\(L(\frac12)\) \(\approx\) \(0.2224121021 + 0.1387832875i\)
\(L(1)\) \(\approx\) \(0.4105362528 + 0.0003494670499i\)
\(L(1)\) \(\approx\) \(0.4105362528 + 0.0003494670499i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.998 - 0.0483i)T \)
3 \( 1 + (-0.607 + 0.794i)T \)
5 \( 1 + (0.0241 - 0.999i)T \)
7 \( 1 + (-0.970 - 0.239i)T \)
13 \( 1 + (-0.970 - 0.239i)T \)
17 \( 1 + (-0.681 - 0.732i)T \)
19 \( 1 + (0.485 + 0.873i)T \)
23 \( 1 + (-0.168 + 0.985i)T \)
29 \( 1 + (-0.354 - 0.935i)T \)
31 \( 1 + (-0.443 + 0.896i)T \)
37 \( 1 + (-0.748 - 0.663i)T \)
41 \( 1 + (-0.906 - 0.421i)T \)
43 \( 1 + (0.0241 - 0.999i)T \)
47 \( 1 + (-0.262 - 0.964i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.681 - 0.732i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.0241 + 0.999i)T \)
71 \( 1 + (0.715 + 0.698i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.958 - 0.285i)T \)
83 \( 1 + (0.995 - 0.0965i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.926 - 0.377i)T \)
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

−20.06914240865495799702523335531, −19.58123284115264138018882606737, −18.97393874824598523844834535401, −18.32670953290957995257425221181, −17.78023918321418045928221050720, −16.92816016310913816136211097140, −16.37476648155944540399325023385, −15.382672014130285078731356936803, −14.75825158800481042864788771633, −13.665160242019834175112151195517, −12.733467589829633801952022725014, −12.05444718346956192134412165497, −11.219180947207416305855077527581, −10.61589770016531492806697513698, −9.82908723436139023570808402432, −9.01761274252752260569101802289, −7.92719206012775087163094529878, −7.18478947663650921820907226045, −6.53343767158076774629447294213, −6.1639533203466754949788397040, −4.9377855903193698032839704471, −3.2866990685023712881554028234, −2.51841530622926233240418490074, −1.781784239067208177819524878564, −0.24380671312785453181963869623, 0.622713914502479687330186934590, 1.9168299814072258995595576366, 3.19891004270463324284515817386, 3.99254928930533202705377634501, 5.22338399960705148875162381644, 5.771808468552607347104793015898, 6.86376927466717578623878322642, 7.5918040235683572225788835437, 8.79741290381765728070107758397, 9.33736029975080667939886894786, 9.96636797797069099475162253942, 10.52992616134719004897339506737, 11.77510111949678857670584149008, 12.06243098361933484104130802310, 12.975245745033019323716818758360, 14.08703877850931133475938954126, 15.44712059746262390636584691641, 15.70510376516321401191542544624, 16.51669488717239615257865062587, 17.02177659516002816460576624532, 17.58695091362598248145054152325, 18.51410461694129728750009295741, 19.535651349554980608029639321461, 20.101575611499963572848684304097, 20.62707826026373208473345341798

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