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Properties

Label 1-2e7-128.3-r1-0-0
Degree $1$
Conductor $128$
Sign $-0.903 + 0.427i$
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  + (0.195 − 0.980i)3-s + (0.831 + 0.555i)5-s + (−0.923 + 0.382i)7-s + (−0.923 − 0.382i)9-s + (−0.980 + 0.195i)11-s + (−0.831 + 0.555i)13-s + (0.707 − 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.555 − 0.831i)19-s + (0.195 + 0.980i)21-s + (0.382 − 0.923i)23-s + (0.382 + 0.923i)25-s + (−0.555 + 0.831i)27-s + (−0.980 − 0.195i)29-s + i·31-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)3-s + (0.831 + 0.555i)5-s + (−0.923 + 0.382i)7-s + (−0.923 − 0.382i)9-s + (−0.980 + 0.195i)11-s + (−0.831 + 0.555i)13-s + (0.707 − 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.555 − 0.831i)19-s + (0.195 + 0.980i)21-s + (0.382 − 0.923i)23-s + (0.382 + 0.923i)25-s + (−0.555 + 0.831i)27-s + (−0.980 − 0.195i)29-s + i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.903 + 0.427i$
Analytic conductor: \(13.7555\)
Root analytic conductor: \(13.7555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (1:\ ),\ -0.903 + 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01080544529 - 0.04811883302i\)
\(L(\frac12)\) \(\approx\) \(0.01080544529 - 0.04811883302i\)
\(L(1)\) \(\approx\) \(0.7767922808 - 0.1533189681i\)
\(L(1)\) \(\approx\) \(0.7767922808 - 0.1533189681i\)

Euler product

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

−29.02017203467937072578445278134, −28.19183461766363769000952948555, −27.028613276914866863860806505501, −26.052903969312827727391541875870, −25.4408605876000289765495163562, −24.16965815054299519109223307438, −22.85163214073816866663220944886, −21.94438066630307312233991445968, −21.052569665711030178891342336807, −20.19876549067778181117227378467, −19.17625910602973887992726744490, −17.53121256893176375686275919095, −16.757945520013799554931102237376, −15.815024268830413512957452187555, −14.75120129928964733846146913005, −13.42143842691047148605180832163, −12.70634099852770614951380242099, −10.83266841342653370668193832086, −9.99576442655416264003420736108, −9.19766524788358540743961693418, −7.8716385130430979473513174697, −6.05400996680210983609807847090, −5.10228337495491986601415805992, −3.71266948628066986234901181701, −2.338463753370043957892717194179, 0.01706515813440124095065512326, 2.19495414139292679362728781624, 2.86815633836055037035558249917, 5.17489897370804059476561365432, 6.52886139346593338818961745484, 7.12678711911024567969418581370, 8.77555851397690725631596600103, 9.792130617867529432723961515416, 11.13648054766156032194957889272, 12.577506280461577643345480447451, 13.24709758175620825681551663904, 14.277121289713562584921037579292, 15.43111972178792811002135381247, 16.89879958991111342195724028341, 17.95631859705302578108105345833, 18.72737030499385115540718633848, 19.583758806867447133906565586231, 20.86473473493541380083404419216, 22.09829221718507166527198890845, 22.88300551882789625704405691188, 24.12933167885514049493312912260, 25.012590427484733440309255789750, 25.96373594780385370388210853842, 26.49853925434799581110915831683, 28.47471816372605845336724628780

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