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Properties

Label 1-1925-1925.1158-r0-0-0
Degree $1$
Conductor $1925$
Sign $-0.165 + 0.986i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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  + (−0.207 + 0.978i)2-s + (−0.743 − 0.669i)3-s + (−0.913 − 0.406i)4-s + (0.809 − 0.587i)6-s + (0.587 − 0.809i)8-s + (0.104 + 0.994i)9-s + (0.406 + 0.913i)12-s + (0.587 + 0.809i)13-s + (0.669 + 0.743i)16-s + (−0.406 + 0.913i)17-s + (−0.994 − 0.104i)18-s + (0.913 − 0.406i)19-s + (0.994 − 0.104i)23-s + (−0.978 + 0.207i)24-s + (−0.913 + 0.406i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.743 − 0.669i)3-s + (−0.913 − 0.406i)4-s + (0.809 − 0.587i)6-s + (0.587 − 0.809i)8-s + (0.104 + 0.994i)9-s + (0.406 + 0.913i)12-s + (0.587 + 0.809i)13-s + (0.669 + 0.743i)16-s + (−0.406 + 0.913i)17-s + (−0.994 − 0.104i)18-s + (0.913 − 0.406i)19-s + (0.994 − 0.104i)23-s + (−0.978 + 0.207i)24-s + (−0.913 + 0.406i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.165 + 0.986i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ -0.165 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5930591227 + 0.7005511888i\)
\(L(\frac12)\) \(\approx\) \(0.5930591227 + 0.7005511888i\)
\(L(1)\) \(\approx\) \(0.6843815442 + 0.2786195190i\)
\(L(1)\) \(\approx\) \(0.6843815442 + 0.2786195190i\)

Euler product

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

−20.16417755605427547834161335461, −19.03164553537632792541906419187, −18.33898471533939055165367162972, −17.717130161402102678898992895404, −17.15062936592878135208489244278, −16.20691658067140976473994078410, −15.64687480989450814342006194742, −14.664912240341621788694687479246, −13.752009768838718733039300997190, −13.03601121763709431874925534344, −12.1426147602507919372594651935, −11.62831403217362684963802115506, −10.80095449584995201849485952145, −10.36795336530379489508843450946, −9.4428476945212178976372996596, −8.94544413709647907291406040399, −7.90946870293224617743992603029, −6.938168303139588225813449050304, −5.75753039925157845015408108658, −5.11283649990055010743830048203, −4.35244066442585452564916861305, −3.397604708547257606660881019672, −2.81271617281075082939198777461, −1.37576060340705916060327364907, −0.518975324282365524352376602, 0.96486413778231264295898441327, 1.713193366246769295660233220807, 3.19826794068349931548864025182, 4.47082729601694272281189929358, 5.00808394085150021134628607413, 5.98821932874462340186428137236, 6.62565818951595170268458858407, 7.09975688391629786859136077374, 8.1622487066914320704115649165, 8.65941697608711487299043499488, 9.70237668975002555678816166883, 10.53306098366223785161051446118, 11.33950389943830627645974933883, 12.14937723567651764187932762221, 13.10117467543252236674254786199, 13.597658846683262072197001603006, 14.29512243018704440559019754994, 15.35747960528216227232464788893, 15.9129031754847578910800651589, 16.74312410992744815028770651211, 17.25153220707885705768462130265, 17.93965686348223256383097806742, 18.60620180610814521385277286638, 19.208776053473210724330737976546, 19.89840449825220258609503134125

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