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Properties

Label 1-896-896.667-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.970 - 0.240i$
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  + (−0.442 − 0.896i)3-s + (0.946 − 0.321i)5-s + (−0.608 + 0.793i)9-s + (−0.0654 − 0.997i)11-s + (0.195 + 0.980i)13-s + (−0.707 − 0.707i)15-s + (0.258 + 0.965i)17-s + (−0.659 − 0.751i)19-s + (0.793 + 0.608i)23-s + (0.793 − 0.608i)25-s + (0.980 + 0.195i)27-s + (−0.831 − 0.555i)29-s + (0.866 + 0.5i)31-s + (−0.866 + 0.5i)33-s + (−0.321 − 0.946i)37-s + ⋯
L(s)  = 1  + (−0.442 − 0.896i)3-s + (0.946 − 0.321i)5-s + (−0.608 + 0.793i)9-s + (−0.0654 − 0.997i)11-s + (0.195 + 0.980i)13-s + (−0.707 − 0.707i)15-s + (0.258 + 0.965i)17-s + (−0.659 − 0.751i)19-s + (0.793 + 0.608i)23-s + (0.793 − 0.608i)25-s + (0.980 + 0.195i)27-s + (−0.831 − 0.555i)29-s + (0.866 + 0.5i)31-s + (−0.866 + 0.5i)33-s + (−0.321 − 0.946i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.970 - 0.240i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.970 - 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1487219843 - 1.216188894i\)
\(L(\frac12)\) \(\approx\) \(0.1487219843 - 1.216188894i\)
\(L(1)\) \(\approx\) \(0.8946652109 - 0.4071436827i\)
\(L(1)\) \(\approx\) \(0.8946652109 - 0.4071436827i\)

Euler product

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

−22.28831301530234207800755259454, −21.216623202767712490377090707795, −20.705957880958231250127049897064, −20.13730145908789576335194426719, −18.65293343889397862439913090797, −18.10947497088739959642442126857, −17.17421086397331476027620907432, −16.80040566038851916205001499935, −15.63575040352683121856467839160, −14.96402703214886904188059678109, −14.33954593005068256535602674574, −13.2226128852339599073958179052, −12.45358801774419121456934443898, −11.43514182014431683892061331610, −10.49115468336233683866397798163, −10.01632788963453681211212162641, −9.3132720275724403869837431658, −8.26556745527747680017001669400, −7.00506564043917229719466842018, −6.18345136977314294983296430052, −5.27637229500435471540095542518, −4.66213676189736837610762857763, −3.38052955512205074318284538384, −2.54323997896828676742986664456, −1.20211916796974938870294120917, 0.281055002150454712699533360707, 1.44332978508051658317133831186, 2.1157214435156711425103135603, 3.36059861233233576452248314247, 4.77571580965010449934273484874, 5.65067229269404629818362400657, 6.35193757950441629247421382243, 7.04344413425175664041040387295, 8.36177147191878557659449520774, 8.84145254491191134935840881050, 10.00682498556045058231013307963, 10.99682621707782438213927511178, 11.607217764768120475260738167905, 12.70401550762909158676265539576, 13.32723779164891549561191296053, 13.86536696306580268899421897526, 14.795702847734377650200784662539, 16.074819710814955248430440663369, 16.94656144762351883352832966454, 17.26759833245788762244942403003, 18.20536539015835634829097109517, 19.097715234115947422911652221173, 19.43278762338571498345246367861, 20.76809348637218839693463123794, 21.54304158991440566074869619511

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