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Properties

Label 1-145-145.99-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: $\chi_{145} (99, \cdot )$
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

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

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