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Properties

Label 1-153-153.101-r1-0-0
Degree $1$
Conductor $153$
Sign $-0.642 - 0.766i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 8-s − 10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 19-s + (−0.5 − 0.866i)20-s + (0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 8-s − 10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 19-s + (−0.5 − 0.866i)20-s + (0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(16.4421\)
Root analytic conductor: \(16.4421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (1:\ ),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4514974841 + 0.9682394792i\)
\(L(\frac12)\) \(\approx\) \(-0.4514974841 + 0.9682394792i\)
\(L(1)\) \(\approx\) \(0.6530277629 + 0.7782481825i\)
\(L(1)\) \(\approx\) \(0.6530277629 + 0.7782481825i\)

Euler product

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

−27.41635874068335201919071283744, −26.51404906432031521164285421271, −24.81717001323145545257247094870, −23.98329877828098446746752530411, −23.16849611557397659224674294243, −22.29961530257072630665916294971, −20.853725743953110334400853062671, −20.32972021601499666560373095335, −19.70230976885873596901334414259, −18.238458522189468570410377717121, −17.32837751768458733596962407798, −15.93545781794019741363381093991, −14.85033599995117713415278440571, −13.7425483503918618422563899337, −12.69010031106958715606833069885, −11.99173927977217309394078077668, −10.69744934646338618845324635191, −9.85127485646586043906189658698, −8.43913227786714237263898310061, −7.25455146453897332458281432565, −5.261011987414222385646666614731, −4.61892045726689751349231998035, −3.3478004350503072710323769606, −1.65376812082491511300553102860, −0.33726595680704290299442097768, 2.551336861484895919108223401778, 3.75764567594535914901425858385, 5.17898302375945605731921127278, 6.207252744896536678305635172221, 7.4521915393850608670044094535, 8.28102127742407328928840100710, 9.59782787567141924867492074920, 11.4004867911584913579066863774, 11.96491848641828346444001060245, 13.556932122623291086483380802970, 14.366232190002432981639369306849, 15.3604933574870907479731713149, 16.027701312367395131277909319830, 17.3579541979060516753760377721, 18.43688471177316786241085071335, 19.09429984272036289563875992873, 20.88196551979893364361732064613, 21.86536153171293538922164775260, 22.42650842182162932792864762711, 23.75683925699461248607868494047, 24.26354313849763900467950758869, 25.40484545763122232593442118996, 26.503885436144962602586619172945, 26.95053656537062749453599490411, 28.21840152725042383956595789917

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