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Properties

Label 1-896-896.59-r0-0-0
Degree $1$
Conductor $896$
Sign $-0.215 - 0.976i$
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  + (−0.0654 + 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

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

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.215 - 0.976i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.215 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1192871014 + 0.1484301606i\)
\(L(\frac12)\) \(\approx\) \(-0.1192871014 + 0.1484301606i\)
\(L(1)\) \(\approx\) \(0.5682083051 + 0.3622359319i\)
\(L(1)\) \(\approx\) \(0.5682083051 + 0.3622359319i\)

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

−21.13452788653435097403058960092, −20.62198647182959087709968699580, −19.48722086473190161421192415122, −19.16298660874311536177323989372, −18.42360237736638561393950913251, −17.234477300477056876609397112037, −16.81085495539246095157890875821, −15.940439808775340376737664374670, −14.88303194458671311971089845647, −14.07141364679860497074299411959, −13.1695003699858869911406441090, −12.428321408897342206428055754203, −11.94363624937294167856550125614, −11.00514878186486208742400252013, −9.96090828574854500246460075100, −8.69599215691632513249016585365, −8.13855549870873702041096036665, −7.48192938270261422713877115887, −6.40345069078980140328771465104, −5.49296800657337071802614051125, −4.638109377000536475575027790519, −3.3523797720943698280078254363, −2.403814875726098051947807861346, −1.13955229954300565806454230023, −0.09070151573943754101891618650, 2.10046604492256571052410285618, 3.085718514307488138203389690, 3.949358102423317327927596350698, 4.78394997098849412105847792988, 5.66993172272127379326718856005, 6.90361745137031870087974059158, 7.6356018063707914194326099930, 8.60099618236151451570622162021, 9.75983141960794778670132288656, 10.21302116623255814927043243741, 11.07870924102888304734714464689, 11.8949729284351889929865933926, 12.65183246111275150085605558220, 14.02923683915602035968874708370, 14.77129457199794660473822863173, 15.26499280565431573058269878655, 15.95361905750817287462197524123, 17.018819770826585363365176506730, 17.5368041192359317826999614904, 18.702315774737590037649844695467, 19.43247525005516306491490444077, 20.16014900397256099771920946578, 21.021376601809230919295563989252, 21.826578387224046760895005395815, 22.37797411373931835319272438294

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