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Properties

Label 1-1536-1536.917-r1-0-0
Degree $1$
Conductor $1536$
Sign $-0.990 + 0.134i$
Analytic cond. $165.066$
Root an. cond. $165.066$
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  + (−0.803 − 0.595i)5-s + (−0.995 − 0.0980i)7-s + (0.671 + 0.740i)11-s + (−0.146 + 0.989i)13-s + (−0.555 + 0.831i)17-s + (−0.514 + 0.857i)19-s + (0.881 − 0.471i)23-s + (0.290 + 0.956i)25-s + (0.998 − 0.0490i)29-s + (0.382 − 0.923i)31-s + (0.740 + 0.671i)35-s + (−0.970 − 0.242i)37-s + (−0.290 + 0.956i)41-s + (0.336 + 0.941i)43-s + (0.980 − 0.195i)47-s + ⋯
L(s)  = 1  + (−0.803 − 0.595i)5-s + (−0.995 − 0.0980i)7-s + (0.671 + 0.740i)11-s + (−0.146 + 0.989i)13-s + (−0.555 + 0.831i)17-s + (−0.514 + 0.857i)19-s + (0.881 − 0.471i)23-s + (0.290 + 0.956i)25-s + (0.998 − 0.0490i)29-s + (0.382 − 0.923i)31-s + (0.740 + 0.671i)35-s + (−0.970 − 0.242i)37-s + (−0.290 + 0.956i)41-s + (0.336 + 0.941i)43-s + (0.980 − 0.195i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $-0.990 + 0.134i$
Analytic conductor: \(165.066\)
Root analytic conductor: \(165.066\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (917, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1536,\ (1:\ ),\ -0.990 + 0.134i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03088534113 + 0.4568983752i\)
\(L(\frac12)\) \(\approx\) \(0.03088534113 + 0.4568983752i\)
\(L(1)\) \(\approx\) \(0.7626084690 + 0.08475234084i\)
\(L(1)\) \(\approx\) \(0.7626084690 + 0.08475234084i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.803 - 0.595i)T \)
7 \( 1 + (-0.995 - 0.0980i)T \)
11 \( 1 + (0.671 + 0.740i)T \)
13 \( 1 + (-0.146 + 0.989i)T \)
17 \( 1 + (-0.555 + 0.831i)T \)
19 \( 1 + (-0.514 + 0.857i)T \)
23 \( 1 + (0.881 - 0.471i)T \)
29 \( 1 + (0.998 - 0.0490i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (-0.970 - 0.242i)T \)
41 \( 1 + (-0.290 + 0.956i)T \)
43 \( 1 + (0.336 + 0.941i)T \)
47 \( 1 + (0.980 - 0.195i)T \)
53 \( 1 + (0.998 + 0.0490i)T \)
59 \( 1 + (-0.146 - 0.989i)T \)
61 \( 1 + (-0.903 + 0.427i)T \)
67 \( 1 + (-0.427 - 0.903i)T \)
71 \( 1 + (-0.634 - 0.773i)T \)
73 \( 1 + (-0.995 + 0.0980i)T \)
79 \( 1 + (-0.195 + 0.980i)T \)
83 \( 1 + (0.970 - 0.242i)T \)
89 \( 1 + (0.881 + 0.471i)T \)
97 \( 1 + (-0.923 - 0.382i)T \)
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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

−19.757679437460344214329605141603, −19.35893215102690349699497608871, −18.767288084227066986334793915037, −17.75940872606201235950540473823, −17.09140596795262416727498439035, −15.982987048777422141326674863729, −15.65187764235693115879209593176, −14.932609338046291366648910316076, −13.88095579818483913700886806444, −13.32002278568644067616025586176, −12.260046677093670856186891897339, −11.761498432547730743394304632500, −10.71659976551756301774271938846, −10.31371716183392156998619327227, −8.9680936214666033044120425200, −8.68398782166231426709516313180, −7.30556874071281252072729339850, −6.91432739486359764552944337075, −6.04049247998712900711949065431, −5.00678978715007276131428677680, −3.95319042657055100724629584872, −3.10722305501013652257200291951, −2.65210171401974647031368901065, −0.86806084078000296807992638845, −0.118344237859957923364474142714, 1.085111334287325123069043205467, 2.125192580879144246940571999899, 3.35362758377328220868316302697, 4.21480974962252419968983035258, 4.63504976363616213313618216205, 6.08089369066707833448220550913, 6.69054190761157398266593340715, 7.491690135340510359471857754839, 8.542595187871675974356053243243, 9.116588840772415544717047255305, 9.9396908816653700921054229454, 10.8231966214297347287794441692, 11.84086747279328156096689629914, 12.37341344484761089745146842835, 12.98882566602949050215326913531, 13.91665788409104693843729023606, 14.94816984784821018334443980104, 15.402209734382617363361811517285, 16.50089695890649848566843874395, 16.72422409659990375427985300485, 17.59498596059885102865522102836, 18.84488474347588406571094988296, 19.29963269155665699540219003544, 19.844130671427034099316655705015, 20.650252335886794019175254216073

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