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Properties

Label 1-2e7-128.53-r0-0-0
Degree $1$
Conductor $128$
Sign $-0.427 + 0.903i$
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  + (−0.980 + 0.195i)3-s + (0.555 + 0.831i)5-s + (−0.923 − 0.382i)7-s + (0.923 − 0.382i)9-s + (−0.195 + 0.980i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)17-s + (−0.831 − 0.555i)19-s + (0.980 + 0.195i)21-s + (0.382 + 0.923i)23-s + (−0.382 + 0.923i)25-s + (−0.831 + 0.555i)27-s + (0.195 + 0.980i)29-s + i·31-s + ⋯
L(s)  = 1  + (−0.980 + 0.195i)3-s + (0.555 + 0.831i)5-s + (−0.923 − 0.382i)7-s + (0.923 − 0.382i)9-s + (−0.195 + 0.980i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)17-s + (−0.831 − 0.555i)19-s + (0.980 + 0.195i)21-s + (0.382 + 0.923i)23-s + (−0.382 + 0.923i)25-s + (−0.831 + 0.555i)27-s + (0.195 + 0.980i)29-s + i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.427 + 0.903i$
Analytic conductor: \(0.594429\)
Root analytic conductor: \(0.594429\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (0:\ ),\ -0.427 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3082366244 + 0.4867588217i\)
\(L(\frac12)\) \(\approx\) \(0.3082366244 + 0.4867588217i\)
\(L(1)\) \(\approx\) \(0.6370917262 + 0.2641528137i\)
\(L(1)\) \(\approx\) \(0.6370917262 + 0.2641528137i\)

Euler product

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

−28.7294123368528920347743866697, −27.67923283769134587853612903195, −26.638004745372988284092786622819, −25.05970115289006868466101440804, −24.63146050422890896511613632880, −23.41505978242055004627486042100, −22.36789408188993345446552499828, −21.6562485502029284404500699001, −20.4537770550386446111102896492, −19.134430397200939073711792344796, −18.20853252338718544606422653312, −16.983165063204948638633725521950, −16.434415675947128357873819926446, −15.32678963109900001320166718662, −13.43058291259518821097585405807, −12.84986562818989199283590955538, −11.81691071341203863788236290404, −10.476130198903139357356314257511, −9.47545367091852971857675008938, −8.13686344315969867948190200223, −6.43581138508619030150573769448, −5.72399418333145488806363884938, −4.52833215686628204669341103946, −2.539538702528139087059678038072, −0.57553559096348513704743622236, 2.04755975747101535275103081163, 3.809095003310904097561415786513, 5.1686114422827373992609752615, 6.68101723811169256451463565271, 6.95667357206944241632262044808, 9.32552464348150212205232851579, 10.211778503482993058123058568341, 11.04005894896637144777293894823, 12.41508892872118286962959158435, 13.33857007838833159706599725978, 14.7917294054642905352280730915, 15.77389869676235776311843763409, 17.046107859255965899403211852832, 17.65147429415911889894123827203, 18.81986493881300869963163241260, 19.86807140437672001071190486219, 21.53789360283352150832363582631, 21.98427091076094037271501643872, 23.09859614392926134995359777549, 23.71329738267995824528724819655, 25.32687978930027316450636751859, 26.17709671792648858741930790975, 27.0418646212201838383873900044, 28.39585697299923903091021432763, 29.045561831076959129740120565250

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