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Properties

Label 1-153-153.13-r0-0-0
Degree $1$
Conductor $153$
Sign $0.0352 - 0.999i$
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 + (0.866 − 0.5i)5-s + (0.866 + 0.5i)7-s − 8-s − i·10-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)13-s + (0.866 − 0.5i)14-s + (−0.5 + 0.866i)16-s − 19-s + (−0.866 − 0.5i)20-s + (0.866 − 0.5i)22-s + (−0.866 + 0.5i)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 + (0.866 − 0.5i)5-s + (0.866 + 0.5i)7-s − 8-s − i·10-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)13-s + (0.866 − 0.5i)14-s + (−0.5 + 0.866i)16-s − 19-s + (−0.866 − 0.5i)20-s + (0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0352 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0352 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.0352 - 0.999i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ 0.0352 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.146451506 - 1.106765027i\)
\(L(\frac12)\) \(\approx\) \(1.146451506 - 1.106765027i\)
\(L(1)\) \(\approx\) \(1.248495463 - 0.7634644349i\)
\(L(1)\) \(\approx\) \(1.248495463 - 0.7634644349i\)

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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.866 + 0.5i)T \)
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 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + iT \)
73 \( 1 + iT \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−28.05133134008901546408291480662, −26.85954065815011878028193405496, −26.29568318200590670729497249711, −25.15900753008376681032993512973, −24.37851561452780860332816345099, −23.53641507287821003644736900671, −22.29207491039483970185468179097, −21.62810503585009632485668841029, −20.76219673312109306748304671915, −19.14177297234515689615339062545, −17.92045964119579219188233643811, −17.173630824895821294324129125490, −16.39457534229773697850409824294, −14.7244416740526992658375500937, −14.33278609853987775345796905817, −13.45194968787029037931086441565, −12.07652302503669711922595705975, −10.878282653618593374579467843808, −9.439539652013933735913880103175, −8.33492796696626468342559379231, −6.999490860293043982952534934532, −6.21598384672766413716125674036, −4.902320097969029936810435215152, −3.7654748110103168191030400737, −2.021926386824463290221516275962, 1.48270202494914654235208049577, 2.4491321036410351159268244072, 4.231409766398614024604502950525, 5.23225584954524587599288918230, 6.23231227301024504500122238819, 8.24279994072022052935556775471, 9.39535702277331714438770003667, 10.2570951527880535005904288257, 11.5874865300104499818153918831, 12.43210163383354500247912016461, 13.436699765821814541794060865429, 14.50612777331179551358019291056, 15.273639964119230087535139459351, 17.194063920183487940479092079889, 17.7698105008625828842044141448, 18.98794502182364563187543779777, 20.183485067918127245513203793476, 20.853957727777886219020741041670, 21.83428912352490160037559999483, 22.49486539332064810511578140246, 23.91285357551771229150831357160, 24.66134291583498511166127540500, 25.59474350954172091978338040373, 27.33686330505112688579999953593, 27.877742710550094792906356340051

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