↑

Properties

Label 1-1375-1375.1098-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.0708 - 0.997i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (−0.368 + 0.929i)2-s + (0.125 − 0.992i)3-s + (−0.728 − 0.684i)4-s + (0.876 + 0.481i)6-s + (−0.951 − 0.309i)7-s + (0.904 − 0.425i)8-s + (−0.968 − 0.248i)9-s + (−0.770 + 0.637i)12-s + (−0.248 + 0.968i)13-s + (0.637 − 0.770i)14-s + (0.0627 + 0.998i)16-s + (0.125 + 0.992i)17-s + (0.587 − 0.809i)18-s + (−0.876 − 0.481i)19-s + (−0.425 + 0.904i)21-s + ⋯
L(s)  = 1  + (−0.368 + 0.929i)2-s + (0.125 − 0.992i)3-s + (−0.728 − 0.684i)4-s + (0.876 + 0.481i)6-s + (−0.951 − 0.309i)7-s + (0.904 − 0.425i)8-s + (−0.968 − 0.248i)9-s + (−0.770 + 0.637i)12-s + (−0.248 + 0.968i)13-s + (0.637 − 0.770i)14-s + (0.0627 + 0.998i)16-s + (0.125 + 0.992i)17-s + (0.587 − 0.809i)18-s + (−0.876 − 0.481i)19-s + (−0.425 + 0.904i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0708 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0708 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.0708 - 0.997i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1098, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.0708 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4484369320 - 0.4177220393i\)
\(L(\frac12)\) \(\approx\) \(0.4484369320 - 0.4177220393i\)
\(L(1)\) \(\approx\) \(0.6702434091 + 0.04562326307i\)
\(L(1)\) \(\approx\) \(0.6702434091 + 0.04562326307i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.368 + 0.929i)T \)
3 \( 1 + (0.125 - 0.992i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.248 + 0.968i)T \)
17 \( 1 + (0.125 + 0.992i)T \)
19 \( 1 + (-0.876 - 0.481i)T \)
23 \( 1 + (0.998 + 0.0627i)T \)
29 \( 1 + (0.187 + 0.982i)T \)
31 \( 1 + (-0.425 - 0.904i)T \)
37 \( 1 + (0.248 - 0.968i)T \)
41 \( 1 + (0.968 + 0.248i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (-0.481 - 0.876i)T \)
53 \( 1 + (0.481 + 0.876i)T \)
59 \( 1 + (-0.535 + 0.844i)T \)
61 \( 1 + (0.535 + 0.844i)T \)
67 \( 1 + (0.125 + 0.992i)T \)
71 \( 1 + (-0.992 - 0.125i)T \)
73 \( 1 + (0.368 - 0.929i)T \)
79 \( 1 + (0.425 - 0.904i)T \)
83 \( 1 + (0.904 - 0.425i)T \)
89 \( 1 + (0.637 - 0.770i)T \)
97 \( 1 + (0.684 - 0.728i)T \)
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

−20.79151551455138705817207261953, −20.15212488109789511359999447372, −19.41809085292183717664120874530, −18.825917458968275016795560075333, −17.83637175309218843818197077333, −17.0387274528462227465860840295, −16.39339371553458268635291691365, −15.597900408312264378667321415284, −14.79385339309802840920029821590, −13.84248095353371945711334385248, −12.98736620785318701344153842131, −12.326137137648898036940041607443, −11.40090653590817362140482302877, −10.64603463353291255720188201796, −9.91846238725644630701905626834, −9.44083919582932200592579915743, −8.60379926722410068453854653384, −7.851463615946263599909901384666, −6.58308968286744065913841072012, −5.39599434727533675514181855530, −4.7139549386089995473199423816, −3.61884645644658906885148351219, −3.03726024861248878594711088977, −2.304636208839495785627992854035, −0.68580586230202326724004505110, 0.205304442433165419501409205392, 1.24146536205546258418789002662, 2.28793281526083388132708070250, 3.56254996878480294619981404769, 4.5525595715249676378985960446, 5.77571180768155374778567945876, 6.407650820258337640073506326788, 7.05053089235673265688145480080, 7.65852611287239895190670222190, 8.83773877902637457981068173327, 9.10276984118552041862418673616, 10.26922199621454384464917573459, 11.08958113117929472541824617660, 12.27682413166576493704198085452, 13.15652461400820923724561585791, 13.3982575085854348339963292055, 14.6401849127817394946893617999, 14.87553209904233966179776192669, 16.172889943714302186530978237102, 16.77187416593274497581659241537, 17.31633832558031532607651055526, 18.217834517514477955266452583739, 18.969898276337524309521680511967, 19.45514867707004158816414150455, 19.96329089737822463594088830859

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