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Properties

Label 1-1925-1925.1061-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.906 + 0.421i$
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  + (−0.978 − 0.207i)2-s + (−0.978 + 0.207i)3-s + (0.913 + 0.406i)4-s + 6-s + (−0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.978 − 0.207i)12-s + (−0.809 − 0.587i)13-s + (0.669 + 0.743i)16-s + (−0.5 − 0.866i)17-s + (−0.978 + 0.207i)18-s + (0.913 − 0.406i)19-s + (0.669 + 0.743i)23-s + (0.913 + 0.406i)24-s + (0.669 + 0.743i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (−0.978 + 0.207i)3-s + (0.913 + 0.406i)4-s + 6-s + (−0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.978 − 0.207i)12-s + (−0.809 − 0.587i)13-s + (0.669 + 0.743i)16-s + (−0.5 − 0.866i)17-s + (−0.978 + 0.207i)18-s + (0.913 − 0.406i)19-s + (0.669 + 0.743i)23-s + (0.913 + 0.406i)24-s + (0.669 + 0.743i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.906 + 0.421i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1061, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.906 + 0.421i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5164448953 + 0.1142426674i\)
\(L(\frac12)\) \(\approx\) \(0.5164448953 + 0.1142426674i\)
\(L(1)\) \(\approx\) \(0.4954828095 + 0.008101669260i\)
\(L(1)\) \(\approx\) \(0.4954828095 + 0.008101669260i\)

Euler product

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

−19.76238469936440292132633472938, −18.98977933833633303561176371664, −18.51636408075403493117747832208, −17.73174175439310146276704728170, −17.00259376447186383750492242250, −16.68715324305923432112390201971, −15.8240895622987906913106322097, −15.10533195376044037087874683643, −14.29320625441236611316923473734, −13.179809666459592851376208739468, −12.24251901779338068421454701125, −11.80527823127333874829097100114, −10.80350464760967398423668116486, −10.47669361147871290974253688186, −9.44467876474337588679334086544, −8.87228181924145005200615312467, −7.593839318271621038981356508552, −7.290018633788750027668532058306, −6.34221480617657995336929915833, −5.693236176361716867529945005754, −4.84733242451156636891996245467, −3.736524206150188197538298319505, −2.30370112004787483891033895449, −1.64181077998173754098578579604, −0.46432926706921408978806975611, 0.675920503057793451966238354169, 1.64261602407891108549901423786, 2.81196674442110684514385656475, 3.64871066030149170992007284365, 4.990717114543083902070677414869, 5.47033027101330676039483624418, 6.67984830566507003123469630082, 7.18204172713106372373271247760, 7.90616731784607083935982808916, 9.21022831704339608693522121307, 9.54428756959040790121745433713, 10.40092715708144599092588513358, 11.2208755405537578574336192161, 11.57209140658543755838428205184, 12.52359601263581440961277217544, 13.075664813092984330179875974573, 14.36477044907685361902952880586, 15.46924643491572774746812161878, 15.750553745364627563941167950959, 16.66376947101504988767456371363, 17.28029221817549687392333292276, 17.81616983368942031154641234698, 18.496247154814901077602872615320, 19.16407957483799074396648658553, 20.278069451378220226122749017092

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