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Properties

Label 1-896-896.181-r1-0-0
Degree $1$
Conductor $896$
Sign $0.740 - 0.671i$
Analytic cond. $96.2885$
Root an. cond. $96.2885$
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)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 + (0.382 + 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 − i·33-s + (0.831 − 0.555i)37-s + ⋯
L(s)  = 1  + (0.980 − 0.195i)3-s + (−0.555 − 0.831i)5-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 + (0.382 + 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 − i·33-s + (0.831 − 0.555i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ 0.740 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.036165523 - 1.171178280i\)
\(L(\frac12)\) \(\approx\) \(3.036165523 - 1.171178280i\)
\(L(1)\) \(\approx\) \(1.523178000 - 0.3098899358i\)
\(L(1)\) \(\approx\) \(1.523178000 - 0.3098899358i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.980 - 0.195i)T \)
5 \( 1 + (-0.555 - 0.831i)T \)
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

−21.55745490615928056012877950025, −21.26969803510338456200353230213, −20.12297936566622546944272826851, −19.45924788552280242278089543890, −18.68633817297331174605156800045, −18.38591806161875696664913126684, −16.89869937394472170107286734165, −16.04193506398994134570081254331, −15.4528833032657916534098791115, −14.53325690870566636962017600814, −13.99341809323699390931734032399, −13.253029197666623978518574064195, −12.090591212688251940026034939146, −11.16894538327075565979371974536, −10.47094356852904065987549170931, −9.544561871521006300781222520038, −8.52421488475486771271412637622, −8.00932621713876746520390626158, −6.995476682161271958363114171044, −6.21706288009723945574208923922, −4.83122229553369485496701797001, −3.75215682414697589582200089625, −3.21215208597018674117438715611, −2.263091981892793174058419968817, −0.90911414785011578139834277114, 0.81277735453037927679500942117, 1.63956830143526973519936361352, 2.9751757071250562885740158161, 3.70576816214344567446709802747, 4.74318378680277096804028572222, 5.59832379481693739618665686784, 7.08387597666139239991930632122, 7.77231815374292419387184733117, 8.30660841600328345435990589348, 9.48151086437010864801296007430, 9.81186964241379548238950163250, 11.218860968058811841074632332273, 12.19139551683735896084720715294, 12.856447600605369737629719299758, 13.49372336550078715529428755323, 14.53439230045391987051712624403, 15.27169613115376041969524462459, 15.9391070819150473593269390671, 16.72653655722390670301436632327, 18.00165320354308421090387983844, 18.41746746748405338403623420397, 19.61370629809372120450777165088, 20.05551042114182095778219563318, 20.73311177837920622309881354540, 21.26784385855433806911162282646

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