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Properties

Label 1-153-153.124-r1-0-0
Degree $1$
Conductor $153$
Sign $0.950 + 0.310i$
Analytic cond. $16.4421$
Root an. cond. $16.4421$
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 + (0.793 − 0.608i)5-s + (0.793 + 0.608i)7-s + (0.707 + 0.707i)8-s + (0.923 − 0.382i)10-s + (0.608 − 0.793i)11-s + (−0.866 − 0.5i)13-s + (0.608 + 0.793i)14-s + (0.5 + 0.866i)16-s + (−0.707 + 0.707i)19-s + (0.991 − 0.130i)20-s + (0.793 − 0.608i)22-s + (0.991 + 0.130i)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 + (0.793 − 0.608i)5-s + (0.793 + 0.608i)7-s + (0.707 + 0.707i)8-s + (0.923 − 0.382i)10-s + (0.608 − 0.793i)11-s + (−0.866 − 0.5i)13-s + (0.608 + 0.793i)14-s + (0.5 + 0.866i)16-s + (−0.707 + 0.707i)19-s + (0.991 − 0.130i)20-s + (0.793 − 0.608i)22-s + (0.991 + 0.130i)23-s + (0.258 − 0.965i)25-s + (−0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.950 + 0.310i$
Analytic conductor: \(16.4421\)
Root analytic conductor: \(16.4421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (1:\ ),\ 0.950 + 0.310i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.150776391 + 0.6601166378i\)
\(L(\frac12)\) \(\approx\) \(4.150776391 + 0.6601166378i\)
\(L(1)\) \(\approx\) \(2.351460983 + 0.2873159067i\)
\(L(1)\) \(\approx\) \(2.351460983 + 0.2873159067i\)

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 \( 1 + (0.793 - 0.608i)T \)
7 \( 1 + (0.793 + 0.608i)T \)
11 \( 1 + (0.608 - 0.793i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.991 + 0.130i)T \)
29 \( 1 + (-0.130 - 0.991i)T \)
31 \( 1 + (0.608 + 0.793i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.130 - 0.991i)T \)
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 \( 1 + (-0.793 - 0.608i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (0.608 - 0.793i)T \)
83 \( 1 + (-0.965 - 0.258i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.130 - 0.991i)T \)
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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

−27.89617548251749707092767129843, −26.661413343719425225280632875069, −25.53703905167257904668684785958, −24.6616645073398051614263233983, −23.69659955323700861292725363156, −22.69878607336835596501002707985, −21.82491662470332489343507422366, −21.03283739560933768767142386254, −20.047896947642455596729016140306, −18.99891198715324206572015855491, −17.566919457418510569575042143996, −16.83526116566747178474162691054, −15.075610247652231376180159208135, −14.54003581023758320651904145620, −13.65692037733423210139403818741, −12.54021580739475118373900707834, −11.319158673574783168336894157677, −10.49104784113941868613981249882, −9.35666896150144661777676103103, −7.305001092203226468883323426877, −6.629906131856530662968372949414, −5.1430474494356372754749493250, −4.21473793790091452909829219602, −2.60306105095364586452631332978, −1.53745195551286061408325416127, 1.54942349916852790955644317406, 2.83800019142022398155055750910, 4.51554569021388658131206593167, 5.43842505131951995599581878298, 6.34655623650817549750957265170, 7.924864688940476151546956225178, 8.9443880530782599143558755413, 10.51505703853935134844029864521, 11.79077834066248164544233706873, 12.5968295141250867809773301623, 13.733869149603247637599249014379, 14.55756155240046917955340667315, 15.53645819912378334155940146591, 16.92341574986288420981424153148, 17.36311558008274104597769453402, 18.98778887991510488621253830924, 20.28460927861055154481371592193, 21.26276805588492891104981871678, 21.73281802947246201750647591018, 22.85739055456536414870019558070, 24.146610256120592771067514824415, 24.793868194032605067517521913057, 25.29855594311635593268071642318, 26.7927826990100928387065979995, 27.88339080461059456461653106013

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