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Properties

Label 1-2e7-128.27-r1-0-0
Degree $1$
Conductor $128$
Sign $-0.671 - 0.740i$
Analytic cond. $13.7555$
Root an. cond. $13.7555$
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  + (−0.555 + 0.831i)3-s + (−0.195 + 0.980i)5-s + (−0.382 + 0.923i)7-s + (−0.382 − 0.923i)9-s + (−0.831 + 0.555i)11-s + (0.195 + 0.980i)13-s + (−0.707 − 0.707i)15-s + (0.707 − 0.707i)17-s + (0.980 − 0.195i)19-s + (−0.555 − 0.831i)21-s + (−0.923 + 0.382i)23-s + (−0.923 − 0.382i)25-s + (0.980 + 0.195i)27-s + (−0.831 − 0.555i)29-s − i·31-s + ⋯
L(s)  = 1  + (−0.555 + 0.831i)3-s + (−0.195 + 0.980i)5-s + (−0.382 + 0.923i)7-s + (−0.382 − 0.923i)9-s + (−0.831 + 0.555i)11-s + (0.195 + 0.980i)13-s + (−0.707 − 0.707i)15-s + (0.707 − 0.707i)17-s + (0.980 − 0.195i)19-s + (−0.555 − 0.831i)21-s + (−0.923 + 0.382i)23-s + (−0.923 − 0.382i)25-s + (0.980 + 0.195i)27-s + (−0.831 − 0.555i)29-s − i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.671 - 0.740i$
Analytic conductor: \(13.7555\)
Root analytic conductor: \(13.7555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (1:\ ),\ -0.671 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2069556167 + 0.4668843903i\)
\(L(\frac12)\) \(\approx\) \(-0.2069556167 + 0.4668843903i\)
\(L(1)\) \(\approx\) \(0.5454592046 + 0.4146599754i\)
\(L(1)\) \(\approx\) \(0.5454592046 + 0.4146599754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.555 + 0.831i)T \)
5 \( 1 + (-0.195 + 0.980i)T \)
7 \( 1 + (-0.382 + 0.923i)T \)
11 \( 1 + (-0.831 + 0.555i)T \)
13 \( 1 + (0.195 + 0.980i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
19 \( 1 + (0.980 - 0.195i)T \)
23 \( 1 + (-0.923 + 0.382i)T \)
29 \( 1 + (-0.831 - 0.555i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.980 + 0.195i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
43 \( 1 + (-0.555 - 0.831i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.831 + 0.555i)T \)
59 \( 1 + (-0.195 + 0.980i)T \)
61 \( 1 + (-0.555 + 0.831i)T \)
67 \( 1 + (0.555 - 0.831i)T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (-0.980 + 0.195i)T \)
89 \( 1 + (0.923 + 0.382i)T \)
97 \( 1 + iT \)
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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

−28.16275885907289577378611558800, −27.03944585295774123460087429707, −25.76524012716996780587387951514, −24.70844124699358851040474985779, −23.76112730711665655803738338403, −23.25750352939043242521730984308, −22.0215438326093893988453686897, −20.5422377222100827855990625009, −19.85534944884975127332253962014, −18.66961639751505625678707641327, −17.61160262062237440749145080216, −16.59919613143530820825210385755, −15.94310053246978262230291531042, −14.043461195564937493286317762817, −13.044822185088313465699925398937, −12.459312102335556967027348041466, −11.08239537032811108190763485915, −10.02444779990164740060650248561, −8.22995676590137235014548661245, −7.594433939757076207600829578221, −6.02481172711258568186547730129, −5.069973917903348380010556788677, −3.38212405173471692562055044137, −1.330861503841677259931691135247, −0.23284405237533793105331900112, 2.50628522269263783133209114275, 3.751717199580445396737746392092, 5.24904240089244549166561671204, 6.28018192990515313654701205982, 7.61822548302670664320672268533, 9.37730587324850214180397600901, 10.058441291829532605065982259276, 11.4222501117424356937494129848, 12.02121921231371804944535229164, 13.759386016052191050738863855537, 15.05454055971110030228155860395, 15.661329908293700694465357438923, 16.685234464145845722658358448626, 18.22909547556090518865627092406, 18.61225078008499907215831056566, 20.21887343916857496011959696244, 21.36449843413104071323744749911, 22.15838655448687653075792691322, 22.91811190923165209644062499700, 23.90053941435194526130286149106, 25.55704951647452931964082789302, 26.21987590271135177142219854642, 27.14687131129864094472101975219, 28.302234130933027053842399812138, 28.857661994031016532860061089025

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