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Properties

Label 1-896-896.139-r0-0-0
Degree $1$
Conductor $896$
Sign $0.671 + 0.740i$
Analytic cond. $4.16100$
Root an. cond. $4.16100$
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

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

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.146953883 + 0.9516792042i\)
\(L(\frac12)\) \(\approx\) \(2.146953883 + 0.9516792042i\)
\(L(1)\) \(\approx\) \(1.588144248 + 0.3015521830i\)
\(L(1)\) \(\approx\) \(1.588144248 + 0.3015521830i\)

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

−21.718330760706443922377987592417, −20.98599145789705065950188594421, −20.301546783354191564346469789167, −19.617981905759179566801617151340, −18.87158723207110404106542824702, −17.85288357112987169661463141890, −17.05954836111780326103683638398, −16.12239940435850186681085511636, −15.55955481533190801426413446861, −14.47925291566277683718418695526, −13.848317916058566199758487735714, −13.06042237059040805353590041932, −12.47879895616205007364964025407, −11.22721276741367768794452488284, −10.11909401303580508689503601097, −9.58798561633372416884635690359, −8.64458491281184746877284366094, −8.05736742174582488071644741069, −7.15268932274274020777635266926, −5.66150085058185702149025509414, −5.210800516200439356495102169, −3.89653473815447852061623218365, −3.11159440134153576840647731188, −2.04805816704029032426249580964, −0.96663233864958072877595802723, 1.526760607456544430265531574312, 2.355476109046353779877522348433, 3.11567074580349211558444422737, 4.1964878500759923675508283268, 5.25295215896306660068101727889, 6.56500325060520561868280064751, 7.17867703367829611487675461859, 7.90445574419361758706241172568, 9.0717811315285303550907258380, 9.91933642684978273658747244976, 10.21215350771176312173172272551, 11.72117282210876519713333292509, 12.37449689025082917475030994358, 13.48107555725522424706365573619, 14.14647856131145612576721058132, 14.64104570903247289554831020314, 15.44985711789713813804042691685, 16.43148702078910188007038216022, 17.47651241864839758199055015661, 18.413228177123463003130299524348, 18.70092776023349491330072777724, 19.70988308452178222584458188569, 20.53030901667818335201141620685, 21.13090381948432573127087049402, 21.98865733537841753691230339848

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