Properties

Label 1-160-160.77-r1-0-0
Degree $1$
Conductor $160$
Sign $0.936 + 0.349i$
Analytic cond. $17.1943$
Root an. cond. $17.1943$
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  + (−0.707 + 0.707i)3-s + 7-s i·9-s + (−0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s i·17-s + (−0.707 − 0.707i)19-s + (−0.707 + 0.707i)21-s + 23-s + (0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s + 31-s i·33-s + (0.707 + 0.707i)37-s i·39-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + 7-s i·9-s + (−0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s i·17-s + (−0.707 − 0.707i)19-s + (−0.707 + 0.707i)21-s + 23-s + (0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s + 31-s i·33-s + (0.707 + 0.707i)37-s i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.936 + 0.349i$
Analytic conductor: \(17.1943\)
Root analytic conductor: \(17.1943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (1:\ ),\ 0.936 + 0.349i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.527970144 + 0.2758541557i\)
\(L(\frac12)\) \(\approx\) \(1.527970144 + 0.2758541557i\)
\(L(1)\) \(\approx\) \(1.002689083 + 0.1546252015i\)
\(L(1)\) \(\approx\) \(1.002689083 + 0.1546252015i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 - T \)
11 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 + iT \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 - iT \)
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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

−27.664615094930445553202529018650, −26.65522801476663821840859114055, −25.41968661455981129353368367898, −24.3592927498104363534937702360, −23.69639507959338529486629086603, −22.9435947003079415072865353532, −21.47392187928887293311865640, −20.98814700479434709532745058889, −19.25833807810771741977372239227, −18.668816475692289398802054833273, −17.582182262481589090752748095586, −16.8264117129988888024938428018, −15.65165118814898671812354610659, −14.28110545019446644632821766239, −13.35305046806719688755091106102, −12.27284826987339203744751403951, −11.1692969287992580826360228171, −10.56466790812501318883212614545, −8.59395254533957905806544522699, −7.83680031951483522793703424835, −6.447077160219922514225430035629, −5.50837800214515280556910960715, −4.2166651080655435999024785555, −2.23498573312062758166216839202, −0.97302451789977960844741867853, 0.89648329450307596259127116507, 2.84648901031551712474610888846, 4.5659406636348025375691058490, 5.14953380470211073413267879187, 6.578858511275719986751653726691, 7.95676758783342040557617091689, 9.20172873701781341564155440234, 10.47946887332516089194948668018, 11.147217461346659789577222290033, 12.253000345491976707664294153611, 13.50163961534912472977300588567, 14.97239673915880059062450741429, 15.522481053549237187019178983916, 16.74256571620367325884459086552, 17.78573142331925772657375111207, 18.308874270851443377614998094829, 20.108630200758548694494542436821, 20.92590928744102980772419442107, 21.61616636720615714478663296407, 22.991861614742412663086471814961, 23.38176027144359572311389788109, 24.68284662396084126382516782502, 25.79860179049397723881819586605, 26.9038666221223198159612066965, 27.68063644020973910529219193740

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