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Properties

Label 1-153-153.22-r1-0-0
Degree $1$
Conductor $153$
Sign $-0.140 + 0.990i$
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  + (−0.258 − 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

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

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.140 + 0.990i$
Analytic conductor: \(16.4421\)
Root analytic conductor: \(16.4421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (1:\ ),\ -0.140 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2577943336 + 0.2969385170i\)
\(L(\frac12)\) \(\approx\) \(0.2577943336 + 0.2969385170i\)
\(L(1)\) \(\approx\) \(0.6771486252 - 0.1664979991i\)
\(L(1)\) \(\approx\) \(0.6771486252 - 0.1664979991i\)

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

−27.70034349549170844731603121107, −26.16511369340197082318856920408, −25.62870895794043436278166684123, −24.44407307509245833369369641378, −23.97987988931141258738426669958, −22.86246029573105552889889020993, −21.5933125220118530038036439176, −20.72180723063583857954434899684, −19.273472741957058066345452983801, −18.32783316964208399954849140660, −17.4918099719670564186032625749, −16.22602466942249834762916180302, −15.74049354350215806771492656471, −14.56731035618589333652887351809, −13.2819339120965837100789111161, −12.549261633069395153650369326231, −10.92967395227054889845939201343, −9.417517972568889747382308136381, −8.712355533413739670104053056693, −7.790218045480273893187394198396, −6.189849254296603781952728232833, −5.343946411731012771974706349264, −4.249796562433557739722835962162, −1.98237478779738545468974626964, −0.15979088309930561003601964839, 1.612994495283556892576472855077, 3.0830124709027120205684624235, 4.00606427531472740621293791350, 5.678917244938334565830864476285, 7.33749173966998484518265027863, 8.250475369678836340977829096702, 9.86408376175126305834728234102, 10.65642902824918603348641222542, 11.247094966717293568480113168389, 12.85078148613012804912933682753, 13.64283192353543130287649454153, 14.64408916173762930204461575463, 16.144784155400599651312045222972, 17.44942583922534755462416452541, 18.24153886287592927040064566553, 19.0489136609588633884115427260, 20.22306581610885564080270875124, 20.995336089252972388520975169907, 22.01073655181261247139014945940, 23.069967848238705733471935939192, 23.63696692694904244823640650336, 25.67691676654084527209865964211, 26.15843933298195072512445439274, 27.163687867055199043770900656594, 27.953491076025633344926658180

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