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Properties

Label 1-925-925.347-r0-0-0
Degree $1$
Conductor $925$
Sign $0.839 - 0.543i$
Analytic cond. $4.29568$
Root an. cond. $4.29568$
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  + (0.104 − 0.994i)2-s + (0.207 − 0.978i)3-s + (−0.978 − 0.207i)4-s + (−0.951 − 0.309i)6-s + (−0.866 − 0.5i)7-s + (−0.309 + 0.951i)8-s + (−0.913 − 0.406i)9-s + (0.809 + 0.587i)11-s + (−0.406 + 0.913i)12-s + (0.104 + 0.994i)13-s + (−0.587 + 0.809i)14-s + (0.913 + 0.406i)16-s + (−0.978 + 0.207i)17-s + (−0.5 + 0.866i)18-s + (−0.743 + 0.669i)19-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (0.207 − 0.978i)3-s + (−0.978 − 0.207i)4-s + (−0.951 − 0.309i)6-s + (−0.866 − 0.5i)7-s + (−0.309 + 0.951i)8-s + (−0.913 − 0.406i)9-s + (0.809 + 0.587i)11-s + (−0.406 + 0.913i)12-s + (0.104 + 0.994i)13-s + (−0.587 + 0.809i)14-s + (0.913 + 0.406i)16-s + (−0.978 + 0.207i)17-s + (−0.5 + 0.866i)18-s + (−0.743 + 0.669i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(4.29568\)
Root analytic conductor: \(4.29568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 925,\ (0:\ ),\ 0.839 - 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8660125070 - 0.2557259994i\)
\(L(\frac12)\) \(\approx\) \(0.8660125070 - 0.2557259994i\)
\(L(1)\) \(\approx\) \(0.7183144653 - 0.5146904473i\)
\(L(1)\) \(\approx\) \(0.7183144653 - 0.5146904473i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.743 + 0.669i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.951 - 0.309i)T \)
31 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.743 - 0.669i)T \)
59 \( 1 + (-0.406 + 0.913i)T \)
61 \( 1 + (0.406 + 0.913i)T \)
67 \( 1 + (0.743 - 0.669i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.207 - 0.978i)T \)
83 \( 1 + (0.207 + 0.978i)T \)
89 \( 1 + (0.994 + 0.104i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−21.97429313369131222624122395758, −21.59579882829350088963294165073, −20.3696258764887369814496668149, −19.50346502516681478301577506914, −18.83293609923648761103946156991, −17.60792853169337357443522543169, −17.06141575272669021585884361811, −16.17332286656472036773251890674, −15.558088989952175306577897931149, −15.07574015995970759655031869757, −14.141286189536191794144454834178, −13.3300300677186464455586635707, −12.54914853251860698516472281801, −11.33548743815475406883726523978, −10.34269798997003700805585373506, −9.49656368942234231206327285752, −8.72927769434645736542034928828, −8.32078525009325373318832266816, −6.770154465193829851358944492371, −6.2508503396592986515255313471, −5.23446524416373016292604377561, −4.454913104740127638727051850917, −3.44812251426960445919535181157, −2.72052829024254368349178393491, −0.41732649128205744162575043431, 1.14819339912787303597692275436, 1.93442661705180828247347432523, 2.96432646299771962189065170866, 3.89483862758051160282977608062, 4.75016243729427990825320729021, 6.39450569413059338618677206333, 6.61428989577476543370859933723, 7.95557511218110273699397888302, 8.917042275023814769236647043544, 9.53392215941463817608677895269, 10.51805571400800154518454148066, 11.557054858733449767908810290811, 12.1026364781531096881302885075, 13.01423302482843700045510819785, 13.48561007450545775346671515666, 14.29970427513097945736601418116, 15.09125574988301783879416125376, 16.55411388070960961363221810350, 17.326127234082408186923763723637, 17.95818763319508384944360608909, 19.06934464719334940968941404930, 19.40290115224826646206146019825, 19.95372811179196452988533579062, 20.88700175347102965608337249333, 21.726428987704406159337106417980

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