↑

Properties

Label 1-51e2-2601.256-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.255 + 0.966i$
Analytic cond. $12.0789$
Root an. cond. $12.0789$
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  + (0.816 + 0.577i)2-s + (0.332 + 0.943i)4-s + (−0.696 + 0.717i)5-s + (−0.998 − 0.0615i)7-s + (−0.273 + 0.961i)8-s + (−0.982 + 0.183i)10-s + (0.650 − 0.759i)11-s + (0.969 − 0.243i)13-s + (−0.779 − 0.626i)14-s + (−0.779 + 0.626i)16-s + (0.0922 + 0.995i)19-s + (−0.908 − 0.417i)20-s + (0.969 − 0.243i)22-s + (0.552 − 0.833i)23-s + (−0.0307 − 0.999i)25-s + (0.932 + 0.361i)26-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)2-s + (0.332 + 0.943i)4-s + (−0.696 + 0.717i)5-s + (−0.998 − 0.0615i)7-s + (−0.273 + 0.961i)8-s + (−0.982 + 0.183i)10-s + (0.650 − 0.759i)11-s + (0.969 − 0.243i)13-s + (−0.779 − 0.626i)14-s + (−0.779 + 0.626i)16-s + (0.0922 + 0.995i)19-s + (−0.908 − 0.417i)20-s + (0.969 − 0.243i)22-s + (0.552 − 0.833i)23-s + (−0.0307 − 0.999i)25-s + (0.932 + 0.361i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.255 + 0.966i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.255 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.390125021 + 1.804302521i\)
\(L(\frac12)\) \(\approx\) \(1.390125021 + 1.804302521i\)
\(L(1)\) \(\approx\) \(1.270248427 + 0.7488565317i\)
\(L(1)\) \(\approx\) \(1.270248427 + 0.7488565317i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.816 + 0.577i)T \)
5 \( 1 + (-0.696 + 0.717i)T \)
7 \( 1 + (-0.998 - 0.0615i)T \)
11 \( 1 + (0.650 - 0.759i)T \)
13 \( 1 + (0.969 - 0.243i)T \)
19 \( 1 + (0.0922 + 0.995i)T \)
23 \( 1 + (0.552 - 0.833i)T \)
29 \( 1 + (0.650 - 0.759i)T \)
31 \( 1 + (0.969 - 0.243i)T \)
37 \( 1 + (0.739 - 0.673i)T \)
41 \( 1 + (-0.0307 + 0.999i)T \)
43 \( 1 + (-0.153 + 0.988i)T \)
47 \( 1 + (0.552 + 0.833i)T \)
53 \( 1 + (0.445 + 0.895i)T \)
59 \( 1 + (-0.389 + 0.920i)T \)
61 \( 1 + (-0.389 - 0.920i)T \)
67 \( 1 + (0.816 - 0.577i)T \)
71 \( 1 + (0.445 + 0.895i)T \)
73 \( 1 + (0.932 + 0.361i)T \)
79 \( 1 + (-0.908 - 0.417i)T \)
83 \( 1 + (-0.0307 - 0.999i)T \)
89 \( 1 + (-0.273 - 0.961i)T \)
97 \( 1 + (-0.998 - 0.0615i)T \)
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

−19.373660582579675771727481210568, −18.839397808060960933912767634470, −17.81607126671235931968483466653, −16.85506891476517879631021453293, −16.118992531211797764988885504089, −15.42680041474653660104114269275, −15.141024672634175558931772980160, −13.8078188925312587174354696945, −13.46305293476890194626778182896, −12.605150833888602338392642273733, −12.12834242632033225644236640371, −11.45594748847487364692393273556, −10.69718970541712644531172775163, −9.71512596247991208638004164016, −9.175914419275526266828885737791, −8.41026317530435394689079845506, −6.94798174613686396303741722057, −6.79590403121840926192133444290, −5.628154167266951031111373671048, −4.903129211714562221981659189628, −4.05747743440939135995936522028, −3.544067596798286112881932537505, −2.662002664487863509195865067582, −1.49920503698564426696013239449, −0.70652174340516368183328164729, 0.94396912078400362422630288210, 2.64938908215145632279140631011, 3.13910873221692572132907009052, 3.90549859745898060440448238636, 4.44334658392058958308041978116, 5.930437603915700223428731852162, 6.20764165694853792286636722380, 6.83676286545380064238292659958, 7.856217193528226507475960534765, 8.33623625698843059108307634302, 9.301607512299809627204071240559, 10.3812234382762163638332118556, 11.11814063923179712769502354324, 11.781299641139419745911291292509, 12.517630560600883305228016807813, 13.233057829141835737898351197588, 14.01121178402047101114120925675, 14.51425255378869412632491973888, 15.42593563392699295342871819208, 15.89581371487893074578352914856, 16.526374073406437581484023488380, 17.14207902923585867443702347488, 18.311416593347602208712338326728, 18.795138168134244453439504032652, 19.635760810831934847021080551906

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