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Properties

Label 1-35e2-1225.1133-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.154 - 0.987i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (0.657 − 0.753i)2-s + (−0.351 − 0.936i)3-s + (−0.134 − 0.990i)4-s + (−0.936 − 0.351i)6-s + (−0.834 − 0.550i)8-s + (−0.753 + 0.657i)9-s + (0.753 + 0.657i)11-s + (−0.880 + 0.473i)12-s + (0.919 + 0.393i)13-s + (−0.963 + 0.266i)16-s + (0.990 + 0.134i)17-s + i·18-s + (0.309 + 0.951i)19-s + (0.990 − 0.134i)22-s + (0.880 + 0.473i)23-s + (−0.222 + 0.974i)24-s + ⋯
L(s)  = 1  + (0.657 − 0.753i)2-s + (−0.351 − 0.936i)3-s + (−0.134 − 0.990i)4-s + (−0.936 − 0.351i)6-s + (−0.834 − 0.550i)8-s + (−0.753 + 0.657i)9-s + (0.753 + 0.657i)11-s + (−0.880 + 0.473i)12-s + (0.919 + 0.393i)13-s + (−0.963 + 0.266i)16-s + (0.990 + 0.134i)17-s + i·18-s + (0.309 + 0.951i)19-s + (0.990 − 0.134i)22-s + (0.880 + 0.473i)23-s + (−0.222 + 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.154 - 0.987i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.154 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.366240766 - 1.596550729i\)
\(L(\frac12)\) \(\approx\) \(1.366240766 - 1.596550729i\)
\(L(1)\) \(\approx\) \(1.135617973 - 0.8822573899i\)
\(L(1)\) \(\approx\) \(1.135617973 - 0.8822573899i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.657 - 0.753i)T \)
3 \( 1 + (-0.351 - 0.936i)T \)
11 \( 1 + (0.753 + 0.657i)T \)
13 \( 1 + (0.919 + 0.393i)T \)
17 \( 1 + (0.990 + 0.134i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.880 + 0.473i)T \)
29 \( 1 + (0.691 - 0.722i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.880 - 0.473i)T \)
41 \( 1 + (0.0448 + 0.998i)T \)
43 \( 1 + (0.781 + 0.623i)T \)
47 \( 1 + (0.0896 - 0.995i)T \)
53 \( 1 + (-0.990 + 0.134i)T \)
59 \( 1 + (-0.963 + 0.266i)T \)
61 \( 1 + (-0.983 + 0.178i)T \)
67 \( 1 + (0.951 - 0.309i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (-0.919 + 0.393i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.0896 + 0.995i)T \)
89 \( 1 + (-0.393 - 0.919i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−21.5992751913166280534157757872, −20.79904111619788996835868148139, −20.17873978208711669813110919463, −18.97323807947660990584748135873, −17.93539029096392645675822443133, −17.25700370458337546169225193723, −16.51618751387331727267528771962, −15.948892731261873486292584047668, −15.31167723880424047105159487856, −14.36012546779092063163696733871, −13.95896236109125360816193308816, −12.83288617313601336381870131624, −12.03408154386446092107330014991, −11.19730002569710534961837905241, −10.55091475601274790464186786432, −9.15171178492825067426402884286, −8.85700733811480660264514073202, −7.767736457350673482714066574625, −6.64605105365664058309199114019, −6.00571520032491312587110001461, −5.17589650901925314600019322065, −4.464023109802780989425349244891, −3.38781601027424579070333230856, −3.031771325033727772793447449006, −0.93467321803989556325703295169, 1.075419730573567090485368395814, 1.5900937504503977792831823115, 2.7060651902202212043943347075, 3.70898447301684691151406455140, 4.60887564394717747173888024009, 5.7640389236623156419363203227, 6.18734454358933238340584080789, 7.190286476470612966784016144669, 8.13250375897167892216223250483, 9.27544643992394763019768107860, 10.018696725355620024312037799487, 11.15394376733191493732865515717, 11.58759014909627014054140194609, 12.4201465289372892407254825307, 12.95627018432137610992411510868, 13.88311234943861679885335479326, 14.38737008691635235183626556243, 15.27824354985369087768889511530, 16.430124816284029638996067785068, 17.17438754356117578071406101261, 18.20453951368682447414980999513, 18.691826206324260809571643465295, 19.41613404723179817751977958020, 20.12270391901917352644696625732, 20.939177787034421500627253886230

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