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Properties

Label 1-896-896.269-r1-0-0
Degree $1$
Conductor $896$
Sign $0.891 + 0.453i$
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  + (−0.896 − 0.442i)3-s + (−0.321 + 0.946i)5-s + (0.608 + 0.793i)9-s + (−0.997 − 0.0654i)11-s + (−0.980 − 0.195i)13-s + (0.707 − 0.707i)15-s + (−0.258 + 0.965i)17-s + (−0.751 − 0.659i)19-s + (0.793 − 0.608i)23-s + (−0.793 − 0.608i)25-s + (−0.195 − 0.980i)27-s + (−0.555 − 0.831i)29-s + (0.866 − 0.5i)31-s + (0.866 + 0.5i)33-s + (0.946 + 0.321i)37-s + ⋯
L(s)  = 1  + (−0.896 − 0.442i)3-s + (−0.321 + 0.946i)5-s + (0.608 + 0.793i)9-s + (−0.997 − 0.0654i)11-s + (−0.980 − 0.195i)13-s + (0.707 − 0.707i)15-s + (−0.258 + 0.965i)17-s + (−0.751 − 0.659i)19-s + (0.793 − 0.608i)23-s + (−0.793 − 0.608i)25-s + (−0.195 − 0.980i)27-s + (−0.555 − 0.831i)29-s + (0.866 − 0.5i)31-s + (0.866 + 0.5i)33-s + (0.946 + 0.321i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ 0.891 + 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5793956375 + 0.1388319666i\)
\(L(\frac12)\) \(\approx\) \(0.5793956375 + 0.1388319666i\)
\(L(1)\) \(\approx\) \(0.5941099711 + 0.01034269103i\)
\(L(1)\) \(\approx\) \(0.5941099711 + 0.01034269103i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.896 - 0.442i)T \)
5 \( 1 + (-0.321 + 0.946i)T \)
11 \( 1 + (-0.997 - 0.0654i)T \)
13 \( 1 + (-0.980 - 0.195i)T \)
17 \( 1 + (-0.258 + 0.965i)T \)
19 \( 1 + (-0.751 - 0.659i)T \)
23 \( 1 + (0.793 - 0.608i)T \)
29 \( 1 + (-0.555 - 0.831i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.946 + 0.321i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
43 \( 1 + (-0.831 - 0.555i)T \)
47 \( 1 + (-0.965 + 0.258i)T \)
53 \( 1 + (0.997 + 0.0654i)T \)
59 \( 1 + (0.659 + 0.751i)T \)
61 \( 1 + (-0.0654 - 0.997i)T \)
67 \( 1 + (-0.896 - 0.442i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.991 + 0.130i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 + (0.195 - 0.980i)T \)
89 \( 1 + (-0.130 + 0.991i)T \)
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

−21.51641545631168112120653657592, −21.088882610411837665071121521892, −20.257891279461554701707759949, −19.37748889362400176423161788971, −18.35964907165526294902121945085, −17.644100132793955830499826603169, −16.6705871663927819630178023744, −16.39000578820760485895482835261, −15.42270257136181649176680754716, −14.78382055229803426157873801607, −13.34852560131253069192806739280, −12.7440427459694763783637637612, −11.920169174881309662379729205342, −11.2832212646682058475133636056, −10.22006532475015909001883889681, −9.57478997894818105522904853425, −8.61547370868019042155260195379, −7.581512752635710004860469585036, −6.72878837851682825616903100911, −5.43968659061743737355481860866, −4.99143913903025300223461791252, −4.2308871536747449847063599118, −2.97159725212019915919917680203, −1.54301789310418626647335125852, −0.330250716871626307814366872969, 0.43302582576218655332304386853, 2.08025613770390995386208274101, 2.79602390490066050908652026005, 4.21609759830867917240104328777, 5.07269764145261568866163914020, 6.11373480273878482235609819304, 6.813977388777169866824861464426, 7.613452094872577158021677188418, 8.377584593433260589391168557322, 9.96751233514620064338412388749, 10.50864462987018556125164021797, 11.24143959997909430003967777248, 12.01131120422811239850984507073, 12.98301036689796899647753725957, 13.50065248900890686345978067796, 15.015026282795606443704729214104, 15.14946338157982085175256314875, 16.36190061929300032446126733659, 17.24271779368534298311698100454, 17.75754318121763776518727019815, 18.86936424030560926873938744788, 19.00423133289253090999017792955, 20.08632719280860330770213710699, 21.35856975101815919783434082845, 21.89565338898527026857299650477

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