↑

Properties

Label 1-145-145.3-r0-0-0
Degree $1$
Conductor $145$
Sign $0.973 - 0.227i$
Analytic cond. $0.673377$
Root an. cond. $0.673377$
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  + (0.900 − 0.433i)2-s + (0.623 + 0.781i)3-s + (0.623 − 0.781i)4-s + (0.900 + 0.433i)6-s + (0.781 − 0.623i)7-s + (0.222 − 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.974 + 0.222i)11-s + 12-s + (−0.974 + 0.222i)13-s + (0.433 − 0.900i)14-s + (−0.222 − 0.974i)16-s − 17-s + (0.222 + 0.974i)18-s + (0.781 + 0.623i)19-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 + 0.781i)3-s + (0.623 − 0.781i)4-s + (0.900 + 0.433i)6-s + (0.781 − 0.623i)7-s + (0.222 − 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.974 + 0.222i)11-s + 12-s + (−0.974 + 0.222i)13-s + (0.433 − 0.900i)14-s + (−0.222 − 0.974i)16-s − 17-s + (0.222 + 0.974i)18-s + (0.781 + 0.623i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.973 - 0.227i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.973 - 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.118073278 - 0.2438967544i\)
\(L(\frac12)\) \(\approx\) \(2.118073278 - 0.2438967544i\)
\(L(1)\) \(\approx\) \(1.917849760 - 0.1742915125i\)
\(L(1)\) \(\approx\) \(1.917849760 - 0.1742915125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (0.781 - 0.623i)T \)
11 \( 1 + (-0.974 + 0.222i)T \)
13 \( 1 + (-0.974 + 0.222i)T \)
17 \( 1 - T \)
19 \( 1 + (0.781 + 0.623i)T \)
23 \( 1 + (0.433 - 0.900i)T \)
31 \( 1 + (0.433 + 0.900i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.433 - 0.900i)T \)
59 \( 1 - T \)
61 \( 1 + (0.781 - 0.623i)T \)
67 \( 1 + (-0.974 - 0.222i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (-0.974 - 0.222i)T \)
83 \( 1 + (0.781 + 0.623i)T \)
89 \( 1 + (-0.433 - 0.900i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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

−28.63320605648692725299779622853, −26.88687750783769051643259659849, −26.09299776664855147494341115193, −24.956606560550227466884057840, −24.34540195134715619824410020161, −23.70097462095719706998189779587, −22.392218405212498560526457489328, −21.363614790792865393791869405541, −20.49315500144837264470794848474, −19.44539016214132073843557767590, −18.07876118814410404474018059264, −17.36918665127364856972164914260, −15.64264347861911344915112113042, −15.03698282980314870352676099485, −13.930441975784076152998403002419, −13.122718121510274977040111602916, −12.11931869821929141855138953832, −11.143444759747372973280497079562, −9.12013648258991124410922297932, −7.93279393382400314947874490521, −7.24867196844484523799409525186, −5.801988370422723352132103454122, −4.743650818504740821126000624143, −3.00057079427649731190073273485, −2.11668081154077810011928721581, 1.99172721782615484433609749871, 3.16614257104261049876020078609, 4.57965552069954225461215347438, 5.0725560696967343676655055040, 7.0022856668073000373717281101, 8.24391461566855577167686071662, 9.86892075741164955321890384097, 10.57160005830972876265071916198, 11.651322209862224124048064264692, 13.07122354950799106998664742405, 14.02724569883817942470643805656, 14.80854400672522759547311950631, 15.7120780097008476658722501482, 16.846895128031073547126996662417, 18.41467386581451263778491211690, 19.76895779469701831403301527688, 20.42738168889905737942952498754, 21.19293521176282694443341126684, 22.09413640151872895280180800591, 23.109181090073753415713901947088, 24.24146727684044952353519029507, 25.00839440208609871272156688770, 26.48136171748774016877691000124, 27.074574752919565199104438756582, 28.38236186509975555316039471684

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