↑

Properties

Label 1-1259-1259.1004-r1-0-0
Degree $1$
Conductor $1259$
Sign $-0.888 - 0.458i$
Analytic cond. $135.298$
Root an. cond. $135.298$
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  + (−0.936 + 0.351i)2-s + (−0.136 + 0.990i)3-s + (0.752 − 0.658i)4-s + (−0.586 − 0.809i)5-s + (−0.220 − 0.975i)6-s + (0.877 − 0.478i)7-s + (−0.472 + 0.881i)8-s + (−0.962 − 0.271i)9-s + (0.834 + 0.551i)10-s + (−0.711 − 0.702i)11-s + (0.549 + 0.835i)12-s + (−0.645 − 0.763i)13-s + (−0.653 + 0.757i)14-s + (0.882 − 0.470i)15-s + (0.131 − 0.991i)16-s + (0.316 + 0.948i)17-s + ⋯
L(s)  = 1  + (−0.936 + 0.351i)2-s + (−0.136 + 0.990i)3-s + (0.752 − 0.658i)4-s + (−0.586 − 0.809i)5-s + (−0.220 − 0.975i)6-s + (0.877 − 0.478i)7-s + (−0.472 + 0.881i)8-s + (−0.962 − 0.271i)9-s + (0.834 + 0.551i)10-s + (−0.711 − 0.702i)11-s + (0.549 + 0.835i)12-s + (−0.645 − 0.763i)13-s + (−0.653 + 0.757i)14-s + (0.882 − 0.470i)15-s + (0.131 − 0.991i)16-s + (0.316 + 0.948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.888 - 0.458i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ -0.888 - 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09557612203 - 0.3933008246i\)
\(L(\frac12)\) \(\approx\) \(0.09557612203 - 0.3933008246i\)
\(L(1)\) \(\approx\) \(0.5612929824 + 0.01049971216i\)
\(L(1)\) \(\approx\) \(0.5612929824 + 0.01049971216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.936 + 0.351i)T \)
3 \( 1 + (-0.136 + 0.990i)T \)
5 \( 1 + (-0.586 - 0.809i)T \)
7 \( 1 + (0.877 - 0.478i)T \)
11 \( 1 + (-0.711 - 0.702i)T \)
13 \( 1 + (-0.645 - 0.763i)T \)
17 \( 1 + (0.316 + 0.948i)T \)
19 \( 1 + (0.377 - 0.926i)T \)
23 \( 1 + (-0.930 + 0.365i)T \)
29 \( 1 + (-0.372 - 0.927i)T \)
31 \( 1 + (-0.498 + 0.866i)T \)
37 \( 1 + (0.630 - 0.776i)T \)
41 \( 1 + (0.450 - 0.892i)T \)
43 \( 1 + (0.980 - 0.198i)T \)
47 \( 1 + (-0.171 + 0.985i)T \)
53 \( 1 + (0.971 + 0.237i)T \)
59 \( 1 + (0.553 - 0.832i)T \)
61 \( 1 + (0.704 - 0.709i)T \)
67 \( 1 + (-0.947 - 0.318i)T \)
71 \( 1 + (0.454 - 0.890i)T \)
73 \( 1 + (-0.235 + 0.971i)T \)
79 \( 1 + (0.725 - 0.688i)T \)
83 \( 1 + (0.932 - 0.361i)T \)
89 \( 1 + (0.156 - 0.987i)T \)
97 \( 1 + (0.335 - 0.942i)T \)
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

−20.89272187073957668170711771271, −20.259275774431667938245490228474, −19.48313724657280481899841447964, −18.623283022242313756365736630812, −18.24646525090884099039913283272, −17.90116111245859266649836437198, −16.71841237297124810175623264957, −16.10865527764237333563509147893, −14.85839438578444985182582043662, −14.5110158320507680459870384305, −13.303722741409298991864318902254, −12.057652980231137024502414366605, −11.99584672421936900170888770548, −11.18578408749788410367010019025, −10.31617140469639121554573586633, −9.40477562399029630647216417438, −8.306939875267050280078095075813, −7.62890852977301470987937826638, −7.30832713338931841442533261958, −6.326263604560129781792384351567, −5.23489359316994375893878182851, −3.93086122680939429419255581799, −2.547611910146767418802212617810, −2.293441318955007270291038460396, −1.147849564446994470754520001001, 0.15559981504266884021825461147, 0.80032563800080578228442358204, 2.20632478444109740907575708324, 3.44942524923538233195743955529, 4.46956841045273846912006334365, 5.351406015384222154267579438, 5.83001770727326688620713026217, 7.47129487981867780312263663142, 7.95361057689277100572138226583, 8.65084081509643382910166067195, 9.455997955922283464753792169072, 10.37542584696547481597910117333, 10.95737389405942113885455130958, 11.60422934203344219273557316949, 12.58846693943747717407540157250, 13.87388675810115270865676537487, 14.72110318792839590504958798827, 15.47118851807828900300838317941, 15.98953331135723443285464579367, 16.70574787417721465743928598645, 17.477146558115718055346700276282, 17.84362129248175705771037361574, 19.2373479744872562414229821345, 19.794300533849726417779566531811, 20.4828064582417623812105546677

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