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Properties

Label 1-145-145.57-r1-0-0
Degree $1$
Conductor $145$
Sign $-0.850 - 0.525i$
Analytic cond. $15.5824$
Root an. cond. $15.5824$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + i·3-s − 4-s + 6-s + i·7-s + i·8-s − 9-s − 11-s − i·12-s − i·13-s + 14-s + 16-s − i·17-s + i·18-s + 19-s + ⋯
L(s)  = 1  − i·2-s + i·3-s − 4-s + 6-s + i·7-s + i·8-s − 9-s − 11-s − i·12-s − i·13-s + 14-s + 16-s − i·17-s + i·18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1327780773 - 0.4673983536i\)
\(L(\frac12)\) \(\approx\) \(0.1327780773 - 0.4673983536i\)
\(L(1)\) \(\approx\) \(0.7181825133 - 0.1695398934i\)
\(L(1)\) \(\approx\) \(0.7181825133 - 0.1695398934i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + iT \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 + T \)
19 \( 1 + iT \)
23 \( 1 + iT \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 - iT \)
43 \( 1 - iT \)
47 \( 1 + T \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + iT \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 - iT \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.50848115353290942385531932976, −27.08787866597317488041874979587, −26.10641678245437450658895871999, −25.587503297953535432848239844761, −24.061460265257668265555824120983, −23.87187882769789618288287073048, −22.976200865565071361039252869663, −21.714735993141600231585492641543, −20.24274633732597637545827839535, −19.09903746857010971055706997396, −18.253622117889194392236657977316, −17.26363363655423010310464803561, −16.50373105614836051033003886817, −15.19970240378666719054713922017, −13.87946331213049852836291351652, −13.49241658390150292192675219986, −12.29259016626432357643106475635, −10.77538338935851246254703179215, −9.349716858651454772234956395, −7.989110608410336536275705163857, −7.33976780154230557799289571923, −6.30536869289966290129006390465, −5.07311868305124461884397163197, −3.530957214897630374629223095770, −1.42149042415977711792007498901, 0.193091479238273279490287591756, 2.49213864964311873246965093286, 3.312335818245313523698763724857, 4.95315792226224274779853956672, 5.53981590717756164550826273906, 8.0134591485702585425419823569, 9.064363042206127597179459902515, 9.98604804971970369560354125999, 10.936540857919177431492193759118, 11.94710280664158824333703641040, 13.04222034105384495110720690682, 14.339207226433258900638505618605, 15.37751002639882368492909100118, 16.36346267144356927167549561217, 17.91915421737249826412408787354, 18.48961833304613698058920864476, 19.98625069413349630555384885337, 20.65291023044658844305401056725, 21.550514236807187613059373138, 22.402417711784745664339052279718, 23.1009350254349437185084530877, 24.73570012047173502388604376112, 25.96604711079314446328363090984, 26.89983841097122089759422945065, 27.7143104810890111499065142720

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