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Properties

Label 1-2e7-128.69-r0-0-0
Degree $1$
Conductor $128$
Sign $0.671 + 0.740i$
Analytic cond. $0.594429$
Root an. cond. $0.594429$
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  + (−0.831 − 0.555i)3-s + (−0.980 − 0.195i)5-s + (−0.382 + 0.923i)7-s + (0.382 + 0.923i)9-s + (0.555 + 0.831i)11-s + (0.980 − 0.195i)13-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (0.195 + 0.980i)19-s + (0.831 − 0.555i)21-s + (−0.923 + 0.382i)23-s + (0.923 + 0.382i)25-s + (0.195 − 0.980i)27-s + (−0.555 + 0.831i)29-s + i·31-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)3-s + (−0.980 − 0.195i)5-s + (−0.382 + 0.923i)7-s + (0.382 + 0.923i)9-s + (0.555 + 0.831i)11-s + (0.980 − 0.195i)13-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (0.195 + 0.980i)19-s + (0.831 − 0.555i)21-s + (−0.923 + 0.382i)23-s + (0.923 + 0.382i)25-s + (0.195 − 0.980i)27-s + (−0.555 + 0.831i)29-s + i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(0.594429\)
Root analytic conductor: \(0.594429\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (0:\ ),\ 0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5705729636 + 0.2529176002i\)
\(L(\frac12)\) \(\approx\) \(0.5705729636 + 0.2529176002i\)
\(L(1)\) \(\approx\) \(0.6976426151 + 0.06235495933i\)
\(L(1)\) \(\approx\) \(0.6976426151 + 0.06235495933i\)

Euler product

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

−28.336730512250287394627405071465, −27.81664565432787162302838948791, −26.587743038612730323918972412823, −26.207074472579336238363039873757, −24.24558879705583406424564728913, −23.49496516899416746357633174615, −22.750285639855725544860603990184, −21.77611450575634261921903980608, −20.57778747199566542238476429258, −19.52722707029615052623007869362, −18.49079319859548850314650535305, −17.10050100976301413451689338492, −16.33779123211096050865825424380, −15.53801968850239494225413010737, −14.231009231329941372953344954254, −12.87011964663447448591377372023, −11.510672661819755696445893664281, −10.97937961874155583897847727926, −9.77168956325543050707345743357, −8.30608680528892295148457695616, −6.91233050935855536464811722987, −5.88911634285213075419714890494, −4.143394214901827843255250879826, −3.605266093801941166130619865077, −0.7140732234628200394748478401, 1.50042838544100525252861036693, 3.47111303211322338592640657412, 4.99887080329018367959001748410, 6.16050110473938915286046851287, 7.35294959093049726900898431260, 8.46660191792783481324382963447, 9.95020125366921859980407641928, 11.44367352619588057908616333286, 12.102710194443507951678321447610, 12.88872110936633783841813610803, 14.49685995585264634660855877399, 15.86623868088130510472098215563, 16.3863111599850482719156609161, 17.89553589435387612816161335906, 18.63991165511321091550264844542, 19.587680615445968039975477585060, 20.78952001301146099524759506462, 22.28854046603295475308686383406, 22.89779144190730880975773594280, 23.75916801357984521803666238116, 24.893470225628608174103301998674, 25.65205142360129177192487311609, 27.4588938764957538084999238810, 27.87242623369062871836775329852, 28.76826371817605490473444314993

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