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Properties

Label 1-145-145.104-r1-0-0
Degree $1$
Conductor $145$
Sign $0.981 - 0.189i$
Analytic cond. $15.5824$
Root an. cond. $15.5824$
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  + i·2-s + i·3-s − 4-s − 6-s − 7-s − i·8-s − 9-s − i·11-s − i·12-s + 13-s − i·14-s + 16-s + i·17-s − i·18-s − i·19-s + ⋯
L(s)  = 1  + i·2-s + i·3-s − 4-s − 6-s − 7-s − i·8-s − 9-s − i·11-s − i·12-s + 13-s − i·14-s + 16-s + i·17-s − i·18-s − i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.981 - 0.189i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ 0.981 - 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6013626196 - 0.05737875721i\)
\(L(\frac12)\) \(\approx\) \(0.6013626196 - 0.05737875721i\)
\(L(1)\) \(\approx\) \(0.6110495990 + 0.4137007196i\)
\(L(1)\) \(\approx\) \(0.6110495990 + 0.4137007196i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + iT \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - iT \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 - iT \)
43 \( 1 + T \)
47 \( 1 - iT \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 - iT \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−28.45386961070780677129563846750, −27.29637493049271626970171869535, −25.871228553172691584132484207067, −25.3376760832992066340980155964, −23.75799052352645402950866278626, −22.9084528131134609034152428281, −22.35039633239879777437551685350, −20.68964225691182082475748521940, −20.08361702374539028924298089040, −18.98040755885659382429481260099, −18.33658622967002081661915384570, −17.37703398442553378364838155434, −15.992523730190518081204940585920, −14.28661276719843899189273983772, −13.449953802763415468672087488208, −12.49717975996997600253231482792, −11.8416017736602260615651867039, −10.43857065087127105749215907891, −9.36695183539128813023581093733, −8.17952597444867035019853101884, −6.82771430679154016802662589569, −5.546692651398587474611214905149, −3.81539234292092408273963373814, −2.58811101141193759406313893522, −1.29694840520510349101650289153, 0.253433008145302801767541131606, 3.29405006637795176259247136466, 4.18727939481613758699391922714, 5.73882609084417536632935490785, 6.36917040586217416500616295747, 8.15526412565293202685207923709, 9.03311594699777892216788838249, 10.01867500028713935605246711442, 11.20171223845373893731919111192, 12.97396039911002560386408830896, 13.85839784028354904195020603259, 15.04454899600202196672825618899, 15.98818077322892610609272194346, 16.45570480859425986587371877581, 17.59647028411386697733219030273, 18.8917289288565892263822883352, 19.90141570322824779261301481067, 21.45922056094674017064644646247, 22.06128380849795940969617355649, 23.06215345850861285263116797479, 23.95826572389594234609864538578, 25.26893023618815174353425344192, 26.25762079248848316109964098204, 26.47765428928872909788800962102, 27.97544946293444672998324945519

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