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Properties

Label 1-6025-6025.161-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.929 - 0.368i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (−0.207 − 0.978i)2-s + (−0.866 − 0.5i)3-s + (−0.913 + 0.406i)4-s + (−0.309 + 0.951i)6-s + (0.0523 + 0.998i)7-s + (0.587 + 0.809i)8-s + (0.5 + 0.866i)9-s + (0.544 − 0.838i)11-s + (0.994 + 0.104i)12-s + (−0.933 − 0.358i)13-s + (0.965 − 0.258i)14-s + (0.669 − 0.743i)16-s + (0.156 + 0.987i)17-s + (0.743 − 0.669i)18-s + (0.933 − 0.358i)19-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.866 − 0.5i)3-s + (−0.913 + 0.406i)4-s + (−0.309 + 0.951i)6-s + (0.0523 + 0.998i)7-s + (0.587 + 0.809i)8-s + (0.5 + 0.866i)9-s + (0.544 − 0.838i)11-s + (0.994 + 0.104i)12-s + (−0.933 − 0.358i)13-s + (0.965 − 0.258i)14-s + (0.669 − 0.743i)16-s + (0.156 + 0.987i)17-s + (0.743 − 0.669i)18-s + (0.933 − 0.358i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1563515789 - 0.8184547548i\)
\(L(\frac12)\) \(\approx\) \(0.1563515789 - 0.8184547548i\)
\(L(1)\) \(\approx\) \(0.5617767767 - 0.3899972752i\)
\(L(1)\) \(\approx\) \(0.5617767767 - 0.3899972752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.207 - 0.978i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.0523 + 0.998i)T \)
11 \( 1 + (0.544 - 0.838i)T \)
13 \( 1 + (-0.933 - 0.358i)T \)
17 \( 1 + (0.156 + 0.987i)T \)
19 \( 1 + (0.933 - 0.358i)T \)
23 \( 1 + (-0.891 - 0.453i)T \)
29 \( 1 + (0.406 - 0.913i)T \)
31 \( 1 + (0.777 - 0.629i)T \)
37 \( 1 + (0.544 - 0.838i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (0.891 - 0.453i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (0.994 - 0.104i)T \)
59 \( 1 + (-0.207 - 0.978i)T \)
61 \( 1 + (0.951 - 0.309i)T \)
67 \( 1 + (-0.994 - 0.104i)T \)
71 \( 1 + (-0.358 - 0.933i)T \)
73 \( 1 + (0.156 - 0.987i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.838 + 0.544i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−17.902063032849460948545188106346, −17.2035147394878841834637299912, −16.64383317333761353783894753057, −16.2210222816284786505927750923, −15.56924935221615252233593700225, −14.75968887391685016564295404323, −14.25118135153478416484591477296, −13.65576328123486330519708689987, −12.75767951317784380026292032761, −11.914150366266790027208072109618, −11.56064490372226374846561525746, −10.22527042863624380503119684287, −10.09972319899444306947208043244, −9.52103100682179153843927045126, −8.708389514539513436880125124850, −7.57443775508960734992793156910, −7.18381450351300968566522666942, −6.70049251215135109283792368264, −5.81693227300491255946387126475, −5.06213811420933165898030306346, −4.524709760667271518059122874249, −4.037924954785633655580216726481, −3.038565778149314208807661623732, −1.47554767983433562619000130275, −0.876898540526155367440582364602, 0.36828962129903620776789681874, 1.1106167216498266140823838236, 2.18304189719196172868032252849, 2.511148526701091314608610458130, 3.60947649012724801021135022666, 4.39359846672045279923108644077, 5.221484915134793913671389519113, 5.8111487838211076655801097119, 6.394171455076246909596715115447, 7.60539014669240287121832197126, 8.03379602360224918375149030981, 8.84872540610292531604873236595, 9.574242478337804286567044261294, 10.2615917116799692955791094039, 10.913120379158769468358015835864, 11.6524308133088640648563720993, 12.09722239476601900315523051798, 12.45211346558297523647610139204, 13.30388597389596557470461884949, 13.91226496702475577774974829756, 14.64801682619638196080868079623, 15.5776088827540770629042187187, 16.30784206016205481292040298930, 17.06315900769934323077229732002, 17.573280898639206502403126328946

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