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Properties

Label 1-2e7-128.11-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  + (−0.980 + 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

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

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: $\chi_{128} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (1:\ ),\ -0.903 - 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07358742405 - 0.3276996804i\)
\(L(\frac12)\) \(\approx\) \(0.07358742405 - 0.3276996804i\)
\(L(1)\) \(\approx\) \(0.6332985103 - 0.08404564967i\)
\(L(1)\) \(\approx\) \(0.6332985103 - 0.08404564967i\)

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

−29.14847368386124833429139204150, −27.79336664272745444520243996698, −27.1426358045478894251305489327, −26.26607368565754999759068161058, −24.68698393467572765758602726296, −23.6488776724809045855797571064, −23.24075320140859223085522568982, −21.90816294733769454382970124390, −21.224799217302009544925661881050, −19.63058124176162779506851674122, −18.53211344169040109612277800060, −17.92447932225320499855700739130, −16.63317721345958809245088776578, −15.77640515073488520478272008318, −14.41991195101680400499879008733, −13.42605921407723572289728676487, −11.79181436751786437597431556226, −11.20484402978949623705191294669, −10.41557717746040687213991659961, −8.505400161595774166617853875468, −7.27471097374725584652346238028, −6.33464405892370474498536880722, −4.92766918393817523994741629730, −3.65544332169175959218838246261, −1.63555410785655516892925250664, 0.15498309260501499814899891952, 1.78509843557392234842559649936, 4.217645735515797491555665721942, 4.92677963690438612973496527341, 6.18402093692832643314305556713, 7.740549094349443441236632179264, 8.78865779043008884134965795600, 10.32525689011285762651579883111, 11.32884864927190246862261466946, 12.30956839117337564416652960140, 13.121739294856826796456088418665, 15.16681711109627480739865727917, 15.55874489671091492869006743578, 17.03972227882666565003927066009, 17.609791445820353499239966440519, 18.747563178801270250452844816556, 20.28231653371137633426621688741, 20.95655649567493213273381058543, 22.160851525643108729269862874164, 23.18621470191127521960952247929, 24.01335072211558170944405029317, 24.79325677826059451381150708091, 26.2698222417912357732122205428, 27.66813278022224447779777755926, 27.951255651542789092809116366

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