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Properties

Label 1-152-152.67-r0-0-0
Degree $1$
Conductor $152$
Sign $-0.660 + 0.750i$
Analytic cond. $0.705885$
Root an. cond. $0.705885$
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  + (−0.173 + 0.984i)3-s + (−0.766 − 0.642i)5-s + (0.5 + 0.866i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)17-s + (−0.939 + 0.342i)21-s + (−0.766 + 0.642i)23-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (−0.939 − 0.342i)29-s + (−0.5 − 0.866i)31-s + (−0.766 − 0.642i)33-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)3-s + (−0.766 − 0.642i)5-s + (0.5 + 0.866i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)17-s + (−0.939 + 0.342i)21-s + (−0.766 + 0.642i)23-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (−0.939 − 0.342i)29-s + (−0.5 − 0.866i)31-s + (−0.766 − 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.660 + 0.750i$
Analytic conductor: \(0.705885\)
Root analytic conductor: \(0.705885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (0:\ ),\ -0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2844607228 + 0.6289478285i\)
\(L(\frac12)\) \(\approx\) \(0.2844607228 + 0.6289478285i\)
\(L(1)\) \(\approx\) \(0.6851889875 + 0.3813896966i\)
\(L(1)\) \(\approx\) \(0.6851889875 + 0.3813896966i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−27.54712803196834329347892130965, −26.73594804311627418863055579569, −25.78589691919303100745003109846, −24.431279946821624009810164067910, −23.82696673749662897788377154883, −22.959944072734106276073739135632, −22.08587166243008947853428221108, −20.38437740943021349655078169982, −19.77787753842252641300079570593, −18.55096720411487557570967555387, −17.99122328805575306412249838274, −16.778862801729009263220796960672, −15.59914967123517709052857711600, −14.32093852793836257458934151916, −13.50883295172219232939864689432, −12.38091563392453952888866157224, −11.09225996718588492944853368236, −10.687126945428389093525763190725, −8.5021484255522013642986415065, −7.68548550905726983319116924148, −6.82176467731550008540855269376, −5.48613997904372225617558978192, −3.82208891818167708396205368593, −2.4840525244298866545566604934, −0.59235327155622213783397117722, 2.16428449108429053099592925150, 3.996049262849928181827349096549, 4.73494726948611826442354918148, 5.91078788759503494586102948384, 7.711575382895628619052867485145, 8.82979367455739389885666157126, 9.62803484244746387413392194250, 11.19782903046766918486598448503, 11.773368574601889235043213846137, 13.01600004489559448387892621762, 14.67526078275725414615082646893, 15.409470243192479266467185765525, 16.172249270684763940907488396616, 17.26226707432773471270572995680, 18.38734004133104168976213120160, 19.73346890454375973679293202530, 20.60403713757459315219388444946, 21.44341126801000090783889157206, 22.38927200024383321007518028434, 23.535255475747299256447422341347, 24.31015532510865324596925749409, 25.654840792529696831991130548480, 26.546150990167546664596243000741, 27.61688502671905084053513823163, 28.25302480527657330272365754490

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