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Properties

Label 1-896-896.237-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.336 + 0.941i$
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  + (−0.555 − 0.831i)3-s + (0.195 + 0.980i)5-s + (−0.382 + 0.923i)9-s + (0.831 + 0.555i)11-s + (−0.195 + 0.980i)13-s + (0.707 − 0.707i)15-s + (−0.707 − 0.707i)17-s + (0.980 + 0.195i)19-s + (0.923 + 0.382i)23-s + (−0.923 + 0.382i)25-s + (0.980 − 0.195i)27-s + (−0.831 + 0.555i)29-s − i·31-s − i·33-s + (0.980 − 0.195i)37-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)3-s + (0.195 + 0.980i)5-s + (−0.382 + 0.923i)9-s + (0.831 + 0.555i)11-s + (−0.195 + 0.980i)13-s + (0.707 − 0.707i)15-s + (−0.707 − 0.707i)17-s + (0.980 + 0.195i)19-s + (0.923 + 0.382i)23-s + (−0.923 + 0.382i)25-s + (0.980 − 0.195i)27-s + (−0.831 + 0.555i)29-s − i·31-s − i·33-s + (0.980 − 0.195i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.336 + 0.941i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7939198614 + 1.127279589i\)
\(L(\frac12)\) \(\approx\) \(0.7939198614 + 1.127279589i\)
\(L(1)\) \(\approx\) \(0.9239704362 + 0.1448284301i\)
\(L(1)\) \(\approx\) \(0.9239704362 + 0.1448284301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.555 - 0.831i)T \)
5 \( 1 + (0.195 + 0.980i)T \)
11 \( 1 + (0.831 + 0.555i)T \)
13 \( 1 + (-0.195 + 0.980i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (0.980 + 0.195i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (-0.831 + 0.555i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.980 - 0.195i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (0.555 - 0.831i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (-0.831 - 0.555i)T \)
59 \( 1 + (-0.195 - 0.980i)T \)
61 \( 1 + (0.555 + 0.831i)T \)
67 \( 1 + (-0.555 - 0.831i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (-0.980 - 0.195i)T \)
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

−21.61024535139112325520649734231, −20.658122920730744771946203346039, −20.20725945231191427089558959833, −19.324985307352541706416726436771, −18.089559420332276223544490149219, −17.22374209277577523443725443156, −16.8920251930056474542345315100, −15.98931158387075195206602930111, −15.28447132405302331415321866277, −14.4598157444487967552336675071, −13.29055448317916974484923486550, −12.65269296253953001231978649498, −11.62719306761361197754515892884, −11.05202792297448691313451859524, −9.983163261725437543154652636672, −9.24744373258793232604406513283, −8.62823923910126934419205643491, −7.49714518120298179947045090143, −6.07959042258684340175447029736, −5.667149928716097054642953992355, −4.60123419310497099377274413125, −3.94790995828125675202268789712, −2.78274596706695424937283675985, −1.17778890712713565504383143620, −0.37033136686540384111040059714, 1.18216917542475880855804368927, 2.0890357847288271718662326037, 3.062374637884032803995573408113, 4.34771551160184933623719170955, 5.407472311688221644036367145587, 6.39494536408393107161080112722, 7.08841282241151289525440317758, 7.48815304337370811076954460793, 9.01298414435116338270149667054, 9.69227739999602244683743834443, 10.964131784100640920400610678465, 11.39636127000013852469263520309, 12.19425930857536239621064038901, 13.14705310340177135117306622706, 14.104841850363902906179703326753, 14.45801312987275043735188562700, 15.673733457517450438620474798050, 16.58884997324348019317461125415, 17.45072953009369506430973666544, 18.019906208767565584118610973146, 18.76173300393552917303761606504, 19.42815094152327096955694593790, 20.22914668920498143526351394408, 21.4357699906054871536008034738, 22.31245424964983927282692763318

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