↑

Properties

Label 1-2349-2349.373-r0-0-0
Degree $1$
Conductor $2349$
Sign $-0.256 - 0.966i$
Analytic cond. $10.9087$
Root an. cond. $10.9087$
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  + (0.514 + 0.857i)2-s + (−0.470 + 0.882i)4-s + (0.302 + 0.953i)5-s + (−0.528 + 0.848i)7-s + (−0.998 + 0.0498i)8-s + (−0.661 + 0.749i)10-s + (0.997 − 0.0664i)11-s + (0.986 + 0.165i)13-s + (−0.999 − 0.0166i)14-s + (−0.556 − 0.830i)16-s + (−0.939 − 0.342i)17-s + (−0.853 − 0.521i)19-s + (−0.983 − 0.181i)20-s + (0.570 + 0.821i)22-s + (0.945 − 0.326i)23-s + ⋯
L(s)  = 1  + (0.514 + 0.857i)2-s + (−0.470 + 0.882i)4-s + (0.302 + 0.953i)5-s + (−0.528 + 0.848i)7-s + (−0.998 + 0.0498i)8-s + (−0.661 + 0.749i)10-s + (0.997 − 0.0664i)11-s + (0.986 + 0.165i)13-s + (−0.999 − 0.0166i)14-s + (−0.556 − 0.830i)16-s + (−0.939 − 0.342i)17-s + (−0.853 − 0.521i)19-s + (−0.983 − 0.181i)20-s + (0.570 + 0.821i)22-s + (0.945 − 0.326i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $-0.256 - 0.966i$
Analytic conductor: \(10.9087\)
Root analytic conductor: \(10.9087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (0:\ ),\ -0.256 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5629811789 + 0.7317990783i\)
\(L(\frac12)\) \(\approx\) \(-0.5629811789 + 0.7317990783i\)
\(L(1)\) \(\approx\) \(0.7092522266 + 0.8378122545i\)
\(L(1)\) \(\approx\) \(0.7092522266 + 0.8378122545i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.514 + 0.857i)T \)
5 \( 1 + (0.302 + 0.953i)T \)
7 \( 1 + (-0.528 + 0.848i)T \)
11 \( 1 + (0.997 - 0.0664i)T \)
13 \( 1 + (0.986 + 0.165i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.853 - 0.521i)T \)
23 \( 1 + (0.945 - 0.326i)T \)
31 \( 1 + (-0.886 + 0.463i)T \)
37 \( 1 + (-0.797 - 0.603i)T \)
41 \( 1 + (-0.835 + 0.549i)T \)
43 \( 1 + (-0.976 + 0.214i)T \)
47 \( 1 + (-0.914 + 0.403i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (-0.286 + 0.957i)T \)
61 \( 1 + (-0.470 - 0.882i)T \)
67 \( 1 + (0.238 - 0.971i)T \)
71 \( 1 + (-0.998 - 0.0498i)T \)
73 \( 1 + (0.270 + 0.962i)T \)
79 \( 1 + (-0.349 + 0.936i)T \)
83 \( 1 + (-0.0912 - 0.995i)T \)
89 \( 1 + (-0.583 + 0.811i)T \)
97 \( 1 + (-0.996 - 0.0830i)T \)
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

−19.43519154985756399198008319117, −18.570467113819824245472445133366, −17.57465589392878201014033553512, −17.02295226050274441257638669995, −16.30414714130351294874448428662, −15.361470970414218861259550383271, −14.62766298010754713486556938807, −13.6431909163812017088633395682, −13.27114467540970521782350399544, −12.74208646911811188366942876713, −11.85765944095667553559713315760, −11.11689633545372573405918420604, −10.38478232280942848851778214822, −9.666727243565239717345460497626, −8.87390310771953373334034089445, −8.406166830108128219827018022786, −6.86773025703952267959605291831, −6.31020128376779840564170277621, −5.43330125035464608227504799158, −4.5122808379498565843034937419, −3.888285128121075226716273670744, −3.29143265843643140137847820544, −1.80733730695463870076230723702, −1.41470653316977895546480085715, −0.23651574399976828022859517341, 1.753304635739450349543075266805, 2.81593307550331778348432647066, 3.41307093965755935654308366368, 4.29999230746366388044232351883, 5.285194588456667654533596135795, 6.19152821029273811071808841879, 6.61653987006330159321287878309, 7.09114083353400959531036611056, 8.42899279539142589663665485276, 8.95350519769645920610345048257, 9.54790531446817493501575838679, 10.84213128387910371743864454457, 11.39839674343654055258577686735, 12.29047108428104983143330608323, 13.11765394317865455175168670154, 13.66408344179824377132253268312, 14.48813943475199579731665841263, 15.1047611893741993462372821309, 15.59204687513008940086775690404, 16.43480350988194247653079338029, 17.154565283124025082217686673459, 17.96669046137806182037099329790, 18.47523470751618748410144405093, 19.18000698235054989651797164778, 20.06239403456183872473751383202

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