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Properties

Label 1-145-145.143-r0-0-0
Degree $1$
Conductor $145$
Sign $0.558 + 0.829i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.558 + 0.829i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.558 + 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.112522302 + 0.5925458521i\)
\(L(\frac12)\) \(\approx\) \(1.112522302 + 0.5925458521i\)
\(L(1)\) \(\approx\) \(1.110939246 + 0.4254089176i\)
\(L(1)\) \(\approx\) \(1.110939246 + 0.4254089176i\)

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

−27.950152988971722573019294807501, −27.117067017217311014370603365730, −26.31770879119988542655478745555, −25.62742844333935924396944614445, −23.96346815234922176437808471866, −23.09857833446730247637891947860, −21.52743332491474315237268363789, −21.114410140090515793774885845317, −20.233469878584972504289870174671, −19.24310690221423205184237875226, −18.45552874043187399855734652206, −17.08611271111175662961959557094, −16.046208995615924899397346894396, −14.51385321205128761965051124164, −13.69146765660421742466058024611, −12.910659166216276413789100546248, −11.148794373019891992739774856771, −10.5934490772336328734278673722, −9.35286137992861649273235528712, −8.40584333910279746345059772734, −7.40291316849637464699916376085, −5.06804587067188827806343336648, −3.92297076310303135725013520898, −2.94563600879001597769742833844, −1.42852421242777844052216572048, 1.65121453810244789692544445184, 3.345188107284606615249340464, 5.01223199657953618586023087100, 6.1643576084094688060927646219, 7.58387664355102856216240327212, 8.24909162494879766878038736345, 9.22710843597391512474207286544, 10.425353945150149374258943465476, 12.40527530512254337101750498901, 13.20553787555602752187614392679, 14.497132702371288220857532290356, 15.090168788541775400843904019360, 16.019332327828005306082912179265, 17.49645937852847396421058339514, 18.51895308827118087962714135816, 18.88072739354247486468053953823, 20.46534319428450212910577801297, 21.31877221298341551833917186568, 22.865748716656311601872046909371, 23.67254139385490520929458048774, 24.76974676447039060552578153316, 25.41884903980726893919770521197, 26.02674100579030361786221678972, 27.32935596012251038653605105997, 28.009009483435015086041099019490

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