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Properties

Label 1-145-145.24-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: $\chi_{145} (24, \cdot )$
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

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

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