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Properties

Label 1-896-896.397-r1-0-0
Degree $1$
Conductor $896$
Sign $0.976 + 0.215i$
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  + (0.0654 + 0.997i)3-s + (−0.659 − 0.751i)5-s + (−0.991 + 0.130i)9-s + (0.442 + 0.896i)11-s + (−0.980 − 0.195i)13-s + (0.707 − 0.707i)15-s + (0.965 − 0.258i)17-s + (0.946 − 0.321i)19-s + (0.130 + 0.991i)23-s + (−0.130 + 0.991i)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 + (−0.751 + 0.659i)37-s + ⋯
L(s)  = 1  + (0.0654 + 0.997i)3-s + (−0.659 − 0.751i)5-s + (−0.991 + 0.130i)9-s + (0.442 + 0.896i)11-s + (−0.980 − 0.195i)13-s + (0.707 − 0.707i)15-s + (0.965 − 0.258i)17-s + (0.946 − 0.321i)19-s + (0.130 + 0.991i)23-s + (−0.130 + 0.991i)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 + (−0.751 + 0.659i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.976 + 0.215i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ 0.976 + 0.215i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.404661629 + 0.1529080970i\)
\(L(\frac12)\) \(\approx\) \(1.404661629 + 0.1529080970i\)
\(L(1)\) \(\approx\) \(0.8982096750 + 0.1778321297i\)
\(L(1)\) \(\approx\) \(0.8982096750 + 0.1778321297i\)

Euler product

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

−22.07503560170180738055520200407, −20.82145259098276335073560232498, −19.7810073795796583905067875409, −19.37521258778441563591206153912, −18.53621554661329912444041624814, −18.12377868980545021886423722616, −16.84949850778785783923025693355, −16.41129682149802645474082249699, −15.08064682711295711596220389153, −14.31713795272213756416363796179, −13.98032911028165841461669016999, −12.618743255857160753649289761365, −12.120269210753671297824312716656, −11.314530801878132760695318640818, −10.52632934277366111724622764759, −9.325892622539185056660108436880, −8.32832771514634076879466510742, −7.57065476601379172618275899543, −6.926053823374559405341209167510, −6.06279947338739988598147304584, −5.06259441379807606897794266734, −3.53547042533674734004625984389, −3.03125718934410235593983290628, −1.79813527542193978189910860711, −0.63903306169990537542107838175, 0.48727868607890310158441170890, 1.93761345642512493800839885924, 3.334276274718384295937930647, 3.961719866773258105303774605838, 5.15714555723913540404114172050, 5.29725764138571066936853846587, 7.06423387214878023134931174802, 7.78709853823584653694038806753, 8.7696375111590663906768486476, 9.71494994500817607212575016579, 9.98069490318144548820472802878, 11.55540868302299153052642084207, 11.76848704549166952987459171280, 12.81182224838222212205976872589, 13.88335833668433105967472017802, 14.87016725194353709826383796861, 15.354063496394536682335804895489, 16.162849719459410626140233519352, 17.01560249893474166548216151212, 17.408765550641317001215519017652, 18.796812905230586908358746144710, 19.69771682404121601832680847778, 20.312038963633918982898258765599, 20.77597379832236119867426372278, 21.86312982464545713838241849790

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