↑

Properties

Label 1-425-425.353-r1-0-0
Degree $1$
Conductor $425$
Sign $0.995 - 0.0972i$
Analytic cond. $45.6725$
Root an. cond. $45.6725$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)3-s + (−0.309 − 0.951i)4-s + (−0.951 − 0.309i)6-s − 7-s + (−0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.587 + 0.809i)11-s + (−0.809 + 0.587i)12-s + (−0.587 − 0.809i)13-s + (−0.587 + 0.809i)14-s + (−0.809 + 0.587i)16-s + i·18-s + (0.309 − 0.951i)19-s + (0.309 + 0.951i)21-s + (0.309 + 0.951i)22-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)3-s + (−0.309 − 0.951i)4-s + (−0.951 − 0.309i)6-s − 7-s + (−0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.587 + 0.809i)11-s + (−0.809 + 0.587i)12-s + (−0.587 − 0.809i)13-s + (−0.587 + 0.809i)14-s + (−0.809 + 0.587i)16-s + i·18-s + (0.309 − 0.951i)19-s + (0.309 + 0.951i)21-s + (0.309 + 0.951i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.995 - 0.0972i$
Analytic conductor: \(45.6725\)
Root analytic conductor: \(45.6725\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 425,\ (1:\ ),\ 0.995 - 0.0972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4500733011 + 0.02194091158i\)
\(L(\frac12)\) \(\approx\) \(0.4500733011 + 0.02194091158i\)
\(L(1)\) \(\approx\) \(0.6009830374 - 0.5987297930i\)
\(L(1)\) \(\approx\) \(0.6009830374 - 0.5987297930i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.951 - 0.309i)T \)
31 \( 1 + (0.951 + 0.309i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.824131710423086376049813213066, −23.036699400079535397842434217140, −22.34945244259343853041456589544, −21.55525137567414221331795183234, −21.011110311541229503366429317139, −19.76770798702875407589219904721, −18.68258270547983721171251101117, −17.51397422282677352336633429219, −16.64985797027338347444617132860, −16.030728771436544365155434692589, −15.53259135295470188329847542360, −14.28204914975317022916790763533, −13.74067060431439685966865867133, −12.43645614406310075215371660111, −11.773932037006029334708819933730, −10.45930176977169284127193668762, −9.53708024886243105692489491420, −8.6622741825310777473112474740, −7.49664885020864193979564257607, −6.25263136185545877221058383923, −5.7027532173642968412966227920, −4.5633935770568049953410772384, −3.64388216725298091500483881113, −2.77579180192147097676131215316, −0.12902144228317443371685691574, 0.87194641160390915863338292404, 2.39318562973003324065474403650, 2.931547909966496430974714111297, 4.521734320092875680299035092966, 5.50460996382837858051533178876, 6.46083866812468572849295863852, 7.3614241668187524291443507676, 8.69311585705543120674600286746, 10.00074757159109818888678763871, 10.52985953544509209346303555036, 11.9628973463664825886191660517, 12.34369228897993703180288246890, 13.2399895383343724312448239770, 13.811877419818952515502965093645, 15.0822554529427571138399521735, 15.90502659514249343831390989584, 17.33362819095767608022586300112, 18.06314452233328399114179945008, 18.90623654883541777694261981632, 19.82348413222077592336056752038, 20.162887329350741416265171960312, 21.50845194792144302794849438248, 22.56947777996953731859184845572, 22.80739864154517420931478241648, 23.7757292846667942896268199730

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