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Properties

Label 1-2e7-128.123-r1-0-0
Degree $1$
Conductor $128$
Sign $0.740 - 0.671i$
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  + (−0.831 − 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

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

Invariants

Degree: \(1\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(13.7555\)
Root analytic conductor: \(13.7555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (1:\ ),\ 0.740 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.592692048 - 0.6143691180i\)
\(L(\frac12)\) \(\approx\) \(1.592692048 - 0.6143691180i\)
\(L(1)\) \(\approx\) \(1.087577765 - 0.2271927926i\)
\(L(1)\) \(\approx\) \(1.087577765 - 0.2271927926i\)

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

−28.713317072567632410271861569576, −27.71197085761126651164312967902, −26.920934590372710117875599471298, −25.58348521293070031290196083592, −24.61567880236323824258501249406, −23.73461754067363795237683401117, −22.186603407461173850979632043434, −21.74399748993062647657731808104, −21.0247120919692878401213593733, −19.45940706330913649840271929438, −18.15396667348528795240325682408, −17.332210593056907850361417119846, −16.544536488741088519508196736288, −15.25058156213838587397428692144, −14.30190027433727595960952553237, −12.80101055753304120809414268506, −11.83618025468308182045234107563, −10.71754185647322608868595412709, −9.579307168011385180907608318040, −8.69058518122267686388622995649, −6.71280322229291934397694307113, −5.59255014432010500678688422855, −4.899268815418447178187728165716, −2.98656588591961691161912042351, −1.19189420314436866417885985957, 0.96721859850731199256065827746, 2.24383763922149296594981642349, 4.43796066380839135927288240321, 5.57042851716888700255076718205, 6.84694659600508175754800243938, 7.60468272732112632676878392600, 9.625106956711126184396097291641, 10.402845627750506826319724436772, 11.676409293254160604210662187989, 12.70144166186851094481233249166, 13.85069639040402694788718117898, 14.6752843362255927243963041174, 16.623764571409130349374180180, 17.15270496094417511341786592121, 17.99423373412598534981714976293, 19.07149706513420037991898251148, 20.40146031511511168276194285246, 21.440074555573616554333351946274, 22.652645168096262464131291507343, 23.172270536453684340212098564719, 24.6215051056571290909252951948, 25.06773936049459703343931862065, 26.52625602077006660521650030564, 27.500476171808670142931421170649, 28.62307386804373955099033472158

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