Properties

Label 1-153-153.130-r1-0-0
Degree $1$
Conductor $153$
Sign $-0.0524 + 0.998i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (0.608 − 0.793i)5-s + (0.608 + 0.793i)7-s + (−0.707 + 0.707i)8-s + (−0.382 + 0.923i)10-s + (−0.793 + 0.608i)11-s + (−0.866 + 0.5i)13-s + (−0.793 − 0.608i)14-s + (0.5 − 0.866i)16-s + (0.707 + 0.707i)19-s + (0.130 − 0.991i)20-s + (0.608 − 0.793i)22-s + (0.130 + 0.991i)23-s + (−0.258 − 0.965i)25-s + (0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (0.608 − 0.793i)5-s + (0.608 + 0.793i)7-s + (−0.707 + 0.707i)8-s + (−0.382 + 0.923i)10-s + (−0.793 + 0.608i)11-s + (−0.866 + 0.5i)13-s + (−0.793 − 0.608i)14-s + (0.5 − 0.866i)16-s + (0.707 + 0.707i)19-s + (0.130 − 0.991i)20-s + (0.608 − 0.793i)22-s + (0.130 + 0.991i)23-s + (−0.258 − 0.965i)25-s + (0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0524 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0524 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.0524 + 0.998i$
Analytic conductor: \(16.4421\)
Root analytic conductor: \(16.4421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (1:\ ),\ -0.0524 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6949259062 + 0.7323582460i\)
\(L(\frac12)\) \(\approx\) \(0.6949259062 + 0.7323582460i\)
\(L(1)\) \(\approx\) \(0.7422110670 + 0.1938113578i\)
\(L(1)\) \(\approx\) \(0.7422110670 + 0.1938113578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 + (0.608 + 0.793i)T \)
11 \( 1 + (-0.793 + 0.608i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.130 + 0.991i)T \)
29 \( 1 + (0.991 + 0.130i)T \)
31 \( 1 + (-0.793 - 0.608i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (-0.991 + 0.130i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.965 + 0.258i)T \)
61 \( 1 + (-0.608 - 0.793i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (-0.793 + 0.608i)T \)
83 \( 1 + (0.965 - 0.258i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.991 + 0.130i)T \)
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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

−27.11530614119596609216690866802, −26.76773349997378935094656739981, −25.85805017213841555553061935762, −24.77156106023739981206128522648, −23.83887627613601323462483805712, −22.358840718495103832524248417000, −21.412120200089921518487209423142, −20.51131504002601036414531701893, −19.50393182878101711040680643326, −18.3742216880118520429804811970, −17.707020051622149963394043514156, −16.83240854685135516420869268758, −15.59275365636991657617572222535, −14.41028327956108567351368740678, −13.3076089358593767341215067504, −11.86053403514742741602846422464, −10.552553616468519815215674149460, −10.35504934078243966441342187105, −8.8350233412979806888583768235, −7.60529654106459334906374806219, −6.82125200965514550864296512271, −5.25785867149256300811626415144, −3.27369303322880075436292547753, −2.182799420257388439000187144706, −0.51970311614797170661323337079, 1.44578273745255881547722847872, 2.45978794835728570868889769966, 4.95512553522715836706196413974, 5.73840924800919573005796862549, 7.31220662909244338183974544555, 8.33423383369965168531770134339, 9.36786910172090814553796132244, 10.14090770899420828902911336204, 11.64049927369356018693832133698, 12.49109254710517847696341891621, 14.05948656475005959775461506791, 15.186302392426864739741050645910, 16.13199313288179487625794255489, 17.22582798399488225675934539728, 17.930009246460432997709023083, 18.86866718659772369975403679125, 20.126752677089128070734956916, 20.92882826380477124108285840652, 21.800536836904004369816467608391, 23.550947361901781387603095547287, 24.42170962937320494463896976039, 25.12796259033887379910709895938, 25.97502798255220964781734006474, 27.18583109947936505270069972028, 27.97026765236815705540338212780

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