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Properties

Label 1-153-153.92-r0-0-0
Degree $1$
Conductor $153$
Sign $0.990 - 0.140i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)4-s + (−0.130 + 0.991i)5-s + (0.130 + 0.991i)7-s + (−0.707 + 0.707i)8-s + (0.923 + 0.382i)10-s + (0.991 − 0.130i)11-s + (0.866 + 0.5i)13-s + (0.991 + 0.130i)14-s + (0.5 + 0.866i)16-s + (−0.707 − 0.707i)19-s + (0.608 − 0.793i)20-s + (0.130 − 0.991i)22-s + (0.608 + 0.793i)23-s + (−0.965 − 0.258i)25-s + (0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)4-s + (−0.130 + 0.991i)5-s + (0.130 + 0.991i)7-s + (−0.707 + 0.707i)8-s + (0.923 + 0.382i)10-s + (0.991 − 0.130i)11-s + (0.866 + 0.5i)13-s + (0.991 + 0.130i)14-s + (0.5 + 0.866i)16-s + (−0.707 − 0.707i)19-s + (0.608 − 0.793i)20-s + (0.130 − 0.991i)22-s + (0.608 + 0.793i)23-s + (−0.965 − 0.258i)25-s + (0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.990 - 0.140i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ 0.990 - 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.147653384 - 0.08098304351i\)
\(L(\frac12)\) \(\approx\) \(1.147653384 - 0.08098304351i\)
\(L(1)\) \(\approx\) \(1.092317203 - 0.2040976826i\)
\(L(1)\) \(\approx\) \(1.092317203 - 0.2040976826i\)

Euler product

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

−27.49848496104496651081617855425, −27.27137617156821380726500066970, −25.80508042979293098397530787829, −25.063000956307645452926172749958, −24.12145353144475440170974274655, −23.30579844905003334438147802687, −22.517379623463193209662296944699, −21.09080305679763671298507085913, −20.31570929853818745776322976884, −19.04026331368782019974947845548, −17.596950365258868594139273280068, −16.90223131293501153032432903753, −16.17311213468183136556624954866, −14.96992837874907302670339306135, −13.89405725866131182187034959248, −13.03357232482405062559022035950, −12.02334020999661708984153212206, −10.39194882821499208342184069640, −8.98610217386915709219790473424, −8.212952489213947731361057505999, −6.99674203477614635684653085142, −5.84536756434748043696370081371, −4.50062146288236152353352400951, −3.7550960225552103170153923437, −1.06008812936136749803982930781, 1.74906862504615901959781916373, 2.99254587458413745766494403665, 4.077680249725961084005366496226, 5.636454763285824317703851113398, 6.748177181372509663253500192893, 8.61730793105995490095825423851, 9.41892468973195144085301459840, 10.90050360888923128883473079032, 11.446473859191780800389813853498, 12.52668913461713116305894016050, 13.8336945760324441622545610754, 14.66967782673710761125313247200, 15.6103149029036278180401981824, 17.38514246812800021243615888936, 18.403322034832376722291237690483, 19.06463656305493123028192770130, 19.9575270992782159595906568, 21.51895309563190581546634499361, 21.75375687667309049358100022143, 22.89542216416654007230412560992, 23.730445024768473068048810183915, 25.13989931912278219392790500840, 26.16703794077869418754654627786, 27.36656336632719253210202429907, 27.946338647609316582091780873123

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