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Properties

Label 1-145-145.48-r0-0-0
Degree $1$
Conductor $145$
Sign $0.999 + 0.0130i$
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  + (0.900 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.999 + 0.0130i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.999 + 0.0130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.065539745 + 0.01351273396i\)
\(L(\frac12)\) \(\approx\) \(2.065539745 + 0.01351273396i\)
\(L(1)\) \(\approx\) \(1.872879470 + 0.03337952261i\)
\(L(1)\) \(\approx\) \(1.872879470 + 0.03337952261i\)

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 \( 1 + (0.623 + 0.781i)T \)
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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

−28.15639472039013389396894639148, −27.59890388116893905760368770084, −26.01583416879200908342257391454, −25.32752541147394524037076915534, −24.32618498103704477517610238273, −22.97368673721320788290144447658, −22.03741037725811188078537423337, −21.578479933582726395633181419929, −20.288093124342729774275195783474, −19.65738291179704004943349582205, −18.6835362740984341132772111020, −16.79769202846505353921594511298, −15.577327308443470424667947892479, −15.208953472596984604008596858576, −13.82232312840450523854196617710, −13.15401563761106969344284018240, −11.7344377360319187336213305771, −10.76152241395809977757723938236, −9.551352237006215349234372433493, −8.672284924659620362874531165515, −6.696110711602618349856639110211, −5.621519834129835120442009378546, −4.17750265089564634797549402808, −3.35130305287967353041468661141, −2.08001806280649779496732948911, 1.82753096526146777207337759030, 3.35961454304050028290480858170, 4.23571158708539581981040695271, 6.380612569971357785321629014889, 6.636025092427144983279153228716, 8.04818797816745842120026345028, 9.129397951544260178674081396428, 10.91686101622376922686193997742, 12.26090972325879026637220402711, 13.0429917273340103069414170722, 13.93044694264219171464104240705, 14.730990854205417323276295779326, 15.99371953661020397888149658763, 16.983942608679795399117570915449, 18.17509269596752795020328977256, 19.6001882828830875084380980696, 20.18405831463239175573746921684, 21.33065642813256055922600704072, 22.667785208221777177789825238163, 23.286106707714532137450184931810, 24.32611897782633418278705033713, 25.19779843678913410244491346698, 25.92861339345883027601315220352, 26.79590439666162122019562781692, 28.60270006749210994900333498910

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