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Properties

Label 1-153-153.67-r0-0-0
Degree $1$
Conductor $153$
Sign $0.766 + 0.642i$
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  + (−0.5 + 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

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

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9031455681 + 0.3287181040i\)
\(L(\frac12)\) \(\approx\) \(0.9031455681 + 0.3287181040i\)
\(L(1)\) \(\approx\) \(0.8815390687 + 0.2842049768i\)
\(L(1)\) \(\approx\) \(0.8815390687 + 0.2842049768i\)

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

−28.1238171292711480542334705736, −27.22873941533576458057666079056, −26.03973667546742980375085366650, −25.02859928467438666148906093244, −24.24466253000479415798514532834, −22.628347475776485042948178863943, −21.74495170729764542000850996538, −20.88726909919155807675774711540, −20.13663262454864021345974281816, −18.99599167320944168710502849170, −17.964035803510105591590461246968, −17.187311145254109881942597090188, −16.19110763241407955645082743488, −14.63066578702431667274572406, −13.46779103519218916245164383835, −12.18034654943595663927654354929, −11.87253491578153199731856615407, −10.234730920875291344523375931768, −9.215342232514799749774032790840, −8.6067043650842708388181892568, −7.11851795500568528822201731744, −5.25902748058787459230868747955, −4.31184475612236691939175494388, −2.456580667556206377911718444206, −1.458105420453354992389466787034, 1.24701624348515397263387192529, 3.268455091386506217430128287380, 4.973759291980049768085431123522, 6.13517966943405025064004535392, 7.20034934814158673015632668879, 8.08307087174956572823080330798, 9.54557274684038384095533109697, 10.4227871367068617119679316031, 11.387937320562300497428910945408, 13.45971284588649965859638947978, 14.103166545973051513675970047573, 14.99485595370204784680656278183, 16.17118424315139249383430848650, 17.39625440080433861809251648600, 17.78017235167524124105528250417, 19.052348183247435483812566154197, 19.89419078526107405853503364137, 21.36515182124428943227090319734, 22.521661923926183007914746002945, 23.25670121808869308697878177981, 24.50759870274984736919944882539, 25.097832465881294992651107577482, 26.42733022965085009318312455654, 26.82456934885890817876831192835, 27.72149428162008493257998116346

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