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Properties

Label 1-153-153.146-r0-0-0
Degree $1$
Conductor $153$
Sign $-0.971 - 0.237i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.971 - 0.237i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ -0.971 - 0.237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1158970696 - 0.9633229128i\)
\(L(\frac12)\) \(\approx\) \(0.1158970696 - 0.9633229128i\)
\(L(1)\) \(\approx\) \(0.6495935902 - 0.7538306617i\)
\(L(1)\) \(\approx\) \(0.6495935902 - 0.7538306617i\)

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

−28.20892979650239163410706570419, −27.31914075341807478380169138992, −26.08611221080284882755191228033, −25.68332536389630982055463773091, −24.686423143465011187233094977553, −23.45849035567078870184630771423, −22.76907366825847642450617197195, −21.77547821505725518694686091400, −21.01342527926167696679017048506, −19.07008091325214655941135909080, −18.34634128177354394567847815891, −17.65068679452379660690928551733, −16.094784285414845799600359941815, −15.43284474487396190250517482291, −14.55353100260794556679150368221, −13.44864963237033467254309438786, −12.46730079437599646877013951647, −11.00712788480796207492140480237, −9.757473137051860657469541755233, −8.453170927072270088082383860812, −7.522598882247237965043728202527, −6.11876506392031712029982108949, −5.57444554460130534458128813831, −3.79222238649301516590983234235, −2.574529210037903829203990933053, 0.781003436203099194683687788753, 2.27932875194189079698095025072, 3.967377129670747370245406544019, 4.74821377060494317478281709140, 6.13443151718565600432425770391, 7.97875722808600450159754861721, 9.056262608401568040911887744206, 10.19506436913275939887848365950, 11.08047708455036217033939030400, 12.37390815132285230047338220298, 13.295885129758770633696967312745, 13.85851115960365914096549128283, 15.470050895812016176963794430672, 16.682111249478061703369106320216, 17.67844862591050201031095171688, 18.83035117080496302529729018499, 19.90709524683951474674936292401, 20.72888926414838246357427583254, 21.25109396161887696668719410251, 22.67110056915382121883469295818, 23.618919099210591073807206889056, 24.16235202389307191881526048112, 25.86394148694723189473121151786, 26.67640570028124433278025189734, 27.99468722030533635822635726845

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