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Properties

Label 1-145-145.139-r0-0-0
Degree $1$
Conductor $145$
Sign $-0.353 + 0.935i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.222 + 0.974i)2-s + (0.900 + 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.222 + 0.974i)6-s + (0.900 + 0.433i)7-s + (−0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.623 − 0.781i)11-s − 12-s + (−0.623 + 0.781i)13-s + (−0.222 + 0.974i)14-s + (0.623 − 0.781i)16-s − 17-s + (−0.623 + 0.781i)18-s + (−0.900 + 0.433i)19-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (0.900 + 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.222 + 0.974i)6-s + (0.900 + 0.433i)7-s + (−0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.623 − 0.781i)11-s − 12-s + (−0.623 + 0.781i)13-s + (−0.222 + 0.974i)14-s + (0.623 − 0.781i)16-s − 17-s + (−0.623 + 0.781i)18-s + (−0.900 + 0.433i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.353 + 0.935i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ -0.353 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8758451850 + 1.267462761i\)
\(L(\frac12)\) \(\approx\) \(0.8758451850 + 1.267462761i\)
\(L(1)\) \(\approx\) \(1.107757126 + 0.9066475442i\)
\(L(1)\) \(\approx\) \(1.107757126 + 0.9066475442i\)

Euler product

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

−27.79344561420156923264391101943, −27.19979924044972594638719548981, −26.11527435147802252867556937066, −24.863273083644632448146893249952, −23.97253176640411012238601480848, −22.96356992271320374105475962235, −21.72037567753609703735518465675, −20.80137752133229434530104505626, −19.88340252146759313857789342303, −19.44302166613970106538423491887, −17.872294084036126059314427621947, −17.52853073242911825746021260076, −15.16522204381565686411064265570, −14.57762526645495554192010715650, −13.48548573647110137070573581218, −12.64182644332159753256486045890, −11.51878126045840820446052318242, −10.30694696831340933160763153199, −9.19057080358770003484380394436, −8.17223956792941275251371738290, −6.90060287638427310297403157122, −4.929180527494936696076829114450, −3.90347978729071491551444141695, −2.47771779997369707721377308091, −1.420963649685855563153193893455, 2.25161441111958682458247853699, 3.95980889959667676028266573128, 4.77820230787994404764515715318, 6.27136005646311220414823934477, 7.601701207781681016160408234479, 8.67072296641406020076565722090, 9.21613867714825887326224912535, 10.886918020416448334915111465731, 12.39342199080104716620259732740, 13.72491649996766912967208442174, 14.48785346083923257223264460867, 15.144058420784996956588947881226, 16.31211130140765854815342477590, 17.21548354250777613161990772148, 18.54522821303590227729598849573, 19.4079967513341479085888509980, 20.91832677049375646612977857147, 21.64967927088748602485455134768, 22.49458574855166828037056290646, 24.25142171832863743719032634280, 24.419919235371389294187960342357, 25.52966984974798702588426193368, 26.60713659322035001803883086702, 27.12685001211647299092901298951, 28.09425545621853920648220823449

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