Properties

Label 1-145-145.27-r0-0-0
Degree $1$
Conductor $145$
Sign $-0.492 + 0.870i$
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 + (−0.433 − 0.900i)7-s + (−0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.781 + 0.623i)11-s + 12-s + (−0.781 + 0.623i)13-s + (−0.974 + 0.222i)14-s + (0.623 + 0.781i)16-s − 17-s + (−0.623 − 0.781i)18-s + (−0.433 + 0.900i)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 + (−0.433 − 0.900i)7-s + (−0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.781 + 0.623i)11-s + 12-s + (−0.781 + 0.623i)13-s + (−0.974 + 0.222i)14-s + (0.623 + 0.781i)16-s − 17-s + (−0.623 − 0.781i)18-s + (−0.433 + 0.900i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.492 + 0.870i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ -0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003296360207 + 0.005654153551i\)
\(L(\frac12)\) \(\approx\) \(0.003296360207 + 0.005654153551i\)
\(L(1)\) \(\approx\) \(0.4698207215 - 0.2068819421i\)
\(L(1)\) \(\approx\) \(0.4698207215 - 0.2068819421i\)

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 \( 1 + (-0.433 - 0.900i)T \)
11 \( 1 + (-0.781 + 0.623i)T \)
13 \( 1 + (-0.781 + 0.623i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.433 + 0.900i)T \)
23 \( 1 + (-0.974 + 0.222i)T \)
31 \( 1 + (-0.974 - 0.222i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.974 + 0.222i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.433 - 0.900i)T \)
67 \( 1 + (-0.781 - 0.623i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.222 + 0.974i)T \)
79 \( 1 + (-0.781 - 0.623i)T \)
83 \( 1 + (-0.433 + 0.900i)T \)
89 \( 1 + (0.974 + 0.222i)T \)
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

−28.77239512978149509850129510449, −27.916255824556999249559331942006, −26.852093974739122675796920700575, −25.77657277697414869639332998521, −24.68335725805336028521534789595, −24.10973626886231834669192090283, −23.12181660665849865616667199499, −21.98933853696561688837179992732, −21.751981829693256230563928586194, −19.64179252763063619213611897021, −18.39873199131000442480237065616, −17.89637358653492637122892300226, −16.703286398333307201384054444683, −15.84678189857671809869351866320, −14.99882124064239716070904621585, −13.36407170971656560172700618868, −12.79786068812961872508489788874, −11.64645499557519084027112832359, −10.20104411662812520800072224170, −8.7916966524430211674981220046, −7.64786630793781181171034107875, −6.46351635937988098982367678144, −5.62899495166970051590780103659, −4.646284429396466124149092331094, −2.67230759497871986334108559171, 0.005593578959840346925138511024, 2.01863630420393975823420670240, 3.85475783163396798686705159912, 4.61844825710024065978706110230, 5.92394172774822429548235855965, 7.361645646299927390513966269214, 9.283749731664214647565738099764, 10.20872820709487158067694153920, 10.873230067720517269982636687575, 12.1181087887197165070712177185, 12.89474355921678891875291552315, 14.11886233781630065139339422451, 15.401543277607198021024756278059, 16.67498880640022537041599919229, 17.5988363111998347192975377164, 18.568555093983623783035482886767, 19.8401710785456528755374331454, 20.662038968758855175000283478881, 21.716600548273535484469066505080, 22.51447148681430281862853921136, 23.39162648564561076986406383922, 24.06095094250140900688221424205, 26.084325174217832606106380567539, 26.8646326183065482647446102944, 27.73998536542167441843815925718

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