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Properties

Label 1-1425-1425.338-r1-0-0
Degree $1$
Conductor $1425$
Sign $-0.240 + 0.970i$
Analytic cond. $153.137$
Root an. cond. $153.137$
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  + (0.927 + 0.374i)2-s + (0.719 + 0.694i)4-s + (−0.866 + 0.5i)7-s + (0.406 + 0.913i)8-s + (0.978 + 0.207i)11-s + (0.788 + 0.615i)13-s + (−0.990 + 0.139i)14-s + (0.0348 + 0.999i)16-s + (0.970 − 0.241i)17-s + (0.829 + 0.559i)22-s + (0.469 + 0.882i)23-s + (0.5 + 0.866i)26-s + (−0.970 − 0.241i)28-s + (0.241 − 0.970i)29-s + (0.104 − 0.994i)31-s + (−0.342 + 0.939i)32-s + ⋯
L(s)  = 1  + (0.927 + 0.374i)2-s + (0.719 + 0.694i)4-s + (−0.866 + 0.5i)7-s + (0.406 + 0.913i)8-s + (0.978 + 0.207i)11-s + (0.788 + 0.615i)13-s + (−0.990 + 0.139i)14-s + (0.0348 + 0.999i)16-s + (0.970 − 0.241i)17-s + (0.829 + 0.559i)22-s + (0.469 + 0.882i)23-s + (0.5 + 0.866i)26-s + (−0.970 − 0.241i)28-s + (0.241 − 0.970i)29-s + (0.104 − 0.994i)31-s + (−0.342 + 0.939i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $-0.240 + 0.970i$
Analytic conductor: \(153.137\)
Root analytic conductor: \(153.137\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1425} (338, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1425,\ (1:\ ),\ -0.240 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.876783710 + 3.675577853i\)
\(L(\frac12)\) \(\approx\) \(2.876783710 + 3.675577853i\)
\(L(1)\) \(\approx\) \(1.830723030 + 0.9071367839i\)
\(L(1)\) \(\approx\) \(1.830723030 + 0.9071367839i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.927 + 0.374i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (0.788 + 0.615i)T \)
17 \( 1 + (0.970 - 0.241i)T \)
23 \( 1 + (0.469 + 0.882i)T \)
29 \( 1 + (0.241 - 0.970i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.0348 + 0.999i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.970 - 0.241i)T \)
53 \( 1 + (0.694 - 0.719i)T \)
59 \( 1 + (-0.848 + 0.529i)T \)
61 \( 1 + (-0.882 + 0.469i)T \)
67 \( 1 + (-0.0697 - 0.997i)T \)
71 \( 1 + (0.438 + 0.898i)T \)
73 \( 1 + (0.788 - 0.615i)T \)
79 \( 1 + (0.559 - 0.829i)T \)
83 \( 1 + (0.994 + 0.104i)T \)
89 \( 1 + (-0.0348 + 0.999i)T \)
97 \( 1 + (0.0697 - 0.997i)T \)
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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

−20.32782232803643374674273373282, −19.77571467669965669834119091239, −19.09760650683772413869636593307, −18.327868599665255517341603442, −17.104172081775408732870069537917, −16.38206609742864761386644133986, −15.82374250138459925001578971429, −14.80957303992513001085071810527, −14.17071576925193372459561681298, −13.50832974669592500381586031309, −12.53633205383444271254414206894, −12.29664693759198745707448262484, −10.995156306596030513083008481680, −10.59750868445436836641433613870, −9.67907814063652725691013677546, −8.81280261118257666298042280775, −7.58945746903988859847410855863, −6.65709727414890831432424743102, −6.13667676414708393597486126242, −5.22382653204501717364161598756, −4.12718439172103754492695639548, −3.47435803866374938619627650689, −2.81341018683355011097759906658, −1.37938550071262657380870790007, −0.70606904418143851847891177596, 1.12498764742841336749251211510, 2.287746013027229563961512092919, 3.2843844770764879354265023939, 3.911158923696000744826883760738, 4.85719581809883299517345735186, 6.04602729771458030556103698540, 6.24938664460333538158801976450, 7.27679713765827181880019774575, 8.125403894838817789217029799082, 9.20336107140781850076647005829, 9.79051094728771193557170174416, 11.15290805811463395057612051591, 11.757621690572075057584958922217, 12.38979776339862556118363928187, 13.339316630466072929732407360345, 13.79044619205374685837528816332, 14.84986585051334111761881570491, 15.27789898127803210731868583992, 16.369748392173304780887591278115, 16.58486182576404141805566260247, 17.587827868955115974986823860951, 18.63050811313089875405011991680, 19.399126865700552177280234495868, 20.07073900758626354569788677309, 21.17119171943948677094285401134

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