Properties

Label 1-105-105.38-r1-0-0
Degree $1$
Conductor $105$
Sign $0.813 + 0.581i$
Analytic cond. $11.2838$
Root an. cond. $11.2838$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s i·8-s + (0.5 + 0.866i)11-s i·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s i·22-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)26-s + 29-s + (0.5 + 0.866i)31-s + (0.866 − 0.5i)32-s + 34-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s i·8-s + (0.5 + 0.866i)11-s i·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s i·22-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)26-s + 29-s + (0.5 + 0.866i)31-s + (0.866 − 0.5i)32-s + 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.813 + 0.581i$
Analytic conductor: \(11.2838\)
Root analytic conductor: \(11.2838\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 105,\ (1:\ ),\ 0.813 + 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9627814922 + 0.3086037039i\)
\(L(\frac12)\) \(\approx\) \(0.9627814922 + 0.3086037039i\)
\(L(1)\) \(\approx\) \(0.7598669203 + 0.008769799075i\)
\(L(1)\) \(\approx\) \(0.7598669203 + 0.008769799075i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + iT \)
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

−29.06490993082341868909198829502, −28.31056643161893585527916343241, −27.03033361068190413371345094413, −26.505160921884719109851635668942, −25.270038429522025431868621806358, −24.368210293070398263207030129490, −23.52178434263885797583643547908, −22.10012493336450873553113608460, −20.85995943378496989635583665717, −19.57281905143085665840039199458, −18.891198241868669739163879207228, −17.667197671753166942708064863006, −16.710741351085207103886921399838, −15.780488957520345088551310329, −14.58505839338957233853180210190, −13.511181624682297645329643005422, −11.65250498436462286564549982025, −10.8133713934863976178472766919, −9.30398791532513874994872140242, −8.60229030709830511192760034388, −7.074146408136281702781573802136, −6.17494625695281691109265234437, −4.55649631400347342231553276808, −2.43090943168424051338935901478, −0.656763106144636492254270513824, 1.28545574320467412604837383845, 2.814021635486487853837418196362, 4.344009238310901585269287945343, 6.346832000547836373518674817279, 7.62495496162157788404088425380, 8.75296833606427697630851110535, 9.94537300750917393211273626622, 10.899527875158376994517040206831, 12.19698044363250037505479412661, 13.06968959711227095685536012436, 14.84033213196302041630689640948, 15.91975728076383453758620820007, 17.28402208602199572639927448465, 17.838613327444404705347714105241, 19.20454056702653794192165610150, 20.01005422237504138210518029260, 20.99395131501536992557818102316, 22.123095672788914329762838243593, 23.226138856447726290735585731529, 24.91635464617152454946938763908, 25.42140627891992476643154942547, 26.73563917612246869275601172413, 27.53188900891677466114426735352, 28.45153276131990912215210992429, 29.44500922570862087760624308814

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