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Properties

Label 1-2e7-128.117-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: $\chi_{128} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (0:\ ),\ 0.427 - 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.048778087 - 0.6641313995i\)
\(L(\frac12)\) \(\approx\) \(1.048778087 - 0.6641313995i\)
\(L(1)\) \(\approx\) \(1.153018543 - 0.3759959547i\)
\(L(1)\) \(\approx\) \(1.153018543 - 0.3759959547i\)

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

−28.962783987355289061263981521722, −27.863710568455335878656961234629, −26.658517317644015897833904688799, −26.07480114715469053185477920599, −25.25704155259764732617430569249, −24.06061054093022769075169799948, −22.6758555868218828806356390418, −22.11463057797711343370301938278, −20.68166945896776701585391388282, −19.802197089494736490059164698564, −18.90984725700005438411837899810, −18.12344370098392373400600431312, −16.22405077716931737601122487635, −15.512190314386865617646830520189, −14.57139163092655965938144407685, −13.53180938689004241466483796996, −12.32460365129940023863274448156, −10.99509833866178351493298358684, −9.68209951695783063034102936632, −8.90422159748200774359135212971, −7.35625052365857848919692498805, −6.63254829930448455963564193753, −4.511226101916603062878795951333, −3.338095664476209638016745137834, −2.29056716731056127873125464883, 1.15702390046217654287286227127, 3.20160752172772010627167747380, 3.96094610446231420193048504762, 5.7979311467166684312168286271, 7.29963106899406027405947993167, 8.386325852806540005547196962217, 9.19205355537907309880591402688, 10.51610569831326432357460167125, 12.0979419845791801125924285577, 13.18137648127093584253415930306, 13.769218044160607197700646256008, 15.42494158639886205038980692551, 15.98457951437193703535038507299, 17.250776864604232018230105305694, 18.80666726874142939504888650975, 19.58822216301507281482039989721, 20.28376420660435951561213389187, 21.29900789481258014587116934021, 22.67210194975293825207211895578, 23.81154043712352502913317153548, 24.65963539271576908492097854832, 25.560154694073706345602141200227, 26.63830479062061810444952788288, 27.3828797035248076016937725301, 28.71855665362729578800298085902

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