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Properties

Label 1-1205-1205.1064-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.612 - 0.790i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (0.5 − 0.866i)2-s + (0.978 − 0.207i)3-s + (−0.5 − 0.866i)4-s + (0.309 − 0.951i)6-s + (0.104 + 0.994i)7-s − 8-s + (0.913 − 0.406i)9-s + (−0.5 + 0.866i)11-s + (−0.669 − 0.743i)12-s + (0.978 − 0.207i)13-s + (0.913 + 0.406i)14-s + (−0.5 + 0.866i)16-s + (0.809 − 0.587i)17-s + (0.104 − 0.994i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.978 − 0.207i)3-s + (−0.5 − 0.866i)4-s + (0.309 − 0.951i)6-s + (0.104 + 0.994i)7-s − 8-s + (0.913 − 0.406i)9-s + (−0.5 + 0.866i)11-s + (−0.669 − 0.743i)12-s + (0.978 − 0.207i)13-s + (0.913 + 0.406i)14-s + (−0.5 + 0.866i)16-s + (0.809 − 0.587i)17-s + (0.104 − 0.994i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1064, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.612 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.540909332 - 1.246338415i\)
\(L(\frac12)\) \(\approx\) \(2.540909332 - 1.246338415i\)
\(L(1)\) \(\approx\) \(1.694467320 - 0.7361787571i\)
\(L(1)\) \(\approx\) \(1.694467320 - 0.7361787571i\)

Euler product

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

−21.416946072944367864332860887661, −20.657898681942944346423065700365, −19.908711080694269340099600606770, −18.895964889512357530656212647848, −18.259824175753932976138309943310, −17.199589447838873206813624527761, −16.39903371061597180012956350924, −15.899084088893876470699183194587, −15.05449331274683117267342142094, −14.155805588435392108239872790920, −13.796429519108071537432899271249, −13.11225843059220790915266952763, −12.28056266079702360872673515526, −10.84865481662405360518062361974, −10.33259911284172590250027099572, −8.98623245189249798390022527148, −8.486681397422195124555417713450, −7.77341939178117343455607451414, −6.931637470988892405721739474268, −6.104644150281258263195979468127, −4.9563991075726054462539846438, −4.094432440636924932828411225645, −3.48680750628940741715336131849, −2.58065369435541063214520521969, −0.99989584256421473416095430013, 1.232144328330145233299363873937, 2.10254980533376736792676967801, 2.82237758481916271058927911552, 3.67664557613301653908911489515, 4.58559783081391512483584495532, 5.60721822534945023593529511433, 6.39903305402816703359941241397, 7.81847007867551288946947977554, 8.35947159838646015295582019879, 9.55092117559780450857289683472, 9.75737853702652917461013398164, 10.92395169648883046752514871243, 11.92836969342523754990246124001, 12.50848148140624869116554291326, 13.23268177138652831396181041151, 13.96845317532215818794208510069, 14.800262489342074694605670997328, 15.33625887656088138296268150611, 16.05987690761216039641070200422, 17.68013608540754915866384670360, 18.41592918852821029409050777789, 18.77878264726756908636259294799, 19.6189245297696884650047280638, 20.50580450738219121257504607148, 20.94418095454006478848398289157

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