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Properties

Label 1-153-153.23-r0-0-0
Degree $1$
Conductor $153$
Sign $0.489 - 0.871i$
Analytic cond. $0.710529$
Root an. cond. $0.710529$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + (0.608 − 0.793i)5-s + (−0.608 − 0.793i)7-s + (0.707 − 0.707i)8-s + (0.382 − 0.923i)10-s + (−0.793 + 0.608i)11-s + (−0.866 + 0.5i)13-s + (−0.793 − 0.608i)14-s + (0.5 − 0.866i)16-s + (0.707 + 0.707i)19-s + (0.130 − 0.991i)20-s + (−0.608 + 0.793i)22-s + (0.130 + 0.991i)23-s + (−0.258 − 0.965i)25-s + (−0.707 + 0.707i)26-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + (0.608 − 0.793i)5-s + (−0.608 − 0.793i)7-s + (0.707 − 0.707i)8-s + (0.382 − 0.923i)10-s + (−0.793 + 0.608i)11-s + (−0.866 + 0.5i)13-s + (−0.793 − 0.608i)14-s + (0.5 − 0.866i)16-s + (0.707 + 0.707i)19-s + (0.130 − 0.991i)20-s + (−0.608 + 0.793i)22-s + (0.130 + 0.991i)23-s + (−0.258 − 0.965i)25-s + (−0.707 + 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.489 - 0.871i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ 0.489 - 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.676567599 - 0.9810933415i\)
\(L(\frac12)\) \(\approx\) \(1.676567599 - 0.9810933415i\)
\(L(1)\) \(\approx\) \(1.658780815 - 0.5958656424i\)
\(L(1)\) \(\approx\) \(1.658780815 - 0.5958656424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 + (-0.608 - 0.793i)T \)
11 \( 1 + (-0.793 + 0.608i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.130 + 0.991i)T \)
29 \( 1 + (0.991 + 0.130i)T \)
31 \( 1 + (0.793 + 0.608i)T \)
37 \( 1 + (0.923 - 0.382i)T \)
41 \( 1 + (-0.991 + 0.130i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (0.608 + 0.793i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + (0.793 - 0.608i)T \)
83 \( 1 + (-0.965 + 0.258i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.991 - 0.130i)T \)
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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

−28.70179211611950792398548147794, −26.81610472192529070234947241789, −26.03900820291663544018650035166, −25.10266392587556046201388123369, −24.35441228616177638261690740302, −23.083250175626917110935628856232, −22.14019964767608877705835312745, −21.74390396675058731515090981374, −20.56658430856056746471043623275, −19.27739430142471264111851896090, −18.23382856348918669602862835007, −17.0415326944233934127512142048, −15.788484800163754478186342270769, −15.092396555924141185437230875096, −13.974743408171808004109491393936, −13.10609472626361702102891306965, −12.07008140494648593078843822340, −10.86360576108894904297991443631, −9.776457642537139118196014148253, −8.1354016644782720287163028929, −6.83243203522950272550299561417, −5.92566368226910697250216663830, −4.93737088112240791928087406358, −3.01449290559805806018959784671, −2.55058068640847325757443978773, 1.477784050020156294946687180521, 2.92404501065846236497802404709, 4.4085730391532669451162189312, 5.27834287991799183077867948103, 6.571920651616378189620483475874, 7.71735195223579691138434120823, 9.67871150056388614243000556730, 10.2141556863124422880977983152, 11.80141800266214178836088665829, 12.759321578363276138133217892087, 13.51138972462191915943001367639, 14.43039819630550434527108533303, 15.833627455874564804368463735052, 16.58106051855670202643713947496, 17.73722394051489776356110929950, 19.361954128662639411535320523124, 20.152666610628292915864068217212, 21.01088472671678041109593034682, 21.86541061973260119260182295768, 23.05697726172469753745119682629, 23.72070180087435979560273584676, 24.810148916904659348346157840096, 25.59085913570000059822141619562, 26.79642368020036863910742066027, 28.30308168251853489014472649959

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