Properties

Label 1-896-896.467-r0-0-0
Degree $1$
Conductor $896$
Sign $0.976 + 0.215i$
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  + (0.997 − 0.0654i)3-s + (−0.751 + 0.659i)5-s + (0.991 − 0.130i)9-s + (−0.896 + 0.442i)11-s + (0.195 − 0.980i)13-s + (−0.707 + 0.707i)15-s + (0.965 − 0.258i)17-s + (−0.321 − 0.946i)19-s + (0.130 + 0.991i)23-s + (0.130 − 0.991i)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 + (0.659 + 0.751i)37-s + ⋯
L(s)  = 1  + (0.997 − 0.0654i)3-s + (−0.751 + 0.659i)5-s + (0.991 − 0.130i)9-s + (−0.896 + 0.442i)11-s + (0.195 − 0.980i)13-s + (−0.707 + 0.707i)15-s + (0.965 − 0.258i)17-s + (−0.321 − 0.946i)19-s + (0.130 + 0.991i)23-s + (0.130 − 0.991i)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 + (0.659 + 0.751i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.976 + 0.215i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.976 + 0.215i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.875702486 + 0.2041844752i\)
\(L(\frac12)\) \(\approx\) \(1.875702486 + 0.2041844752i\)
\(L(1)\) \(\approx\) \(1.354815299 + 0.08310619596i\)
\(L(1)\) \(\approx\) \(1.354815299 + 0.08310619596i\)

Euler product

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

−21.486198668501678891492559990313, −21.06932374839831226307118095461, −20.44958411605010796725852548835, −19.464791479529696559569556776989, −18.95430356093231652572038278952, −18.32028125938774241140954682136, −16.86674396178046777249744261580, −16.18814197414245857934601754032, −15.68076986149597084766662580204, −14.55081636657448790709580479694, −14.1166585258832685875447920955, −12.91242918016191564352241335238, −12.5237048516885841473960728221, −11.42124894927530737005697598994, −10.40617386076939639641136751442, −9.55569691620932201602641905341, −8.5220216932911408783429860912, −8.15281964146731230815849197700, −7.31057902176646412414410283420, −6.11523884269675577631300281456, −4.862612341083446504816874311881, −4.10029994309600720839977920109, −3.26050742012827401110406066150, −2.189686553228058740511707642060, −0.98746833326672135657307701482, 1.02162636904296961313368179191, 2.68130224311246083411545665386, 2.94254621247923989624640894572, 4.07909206620764751223148899169, 5.024034909154488324256282507035, 6.34110094837483973287259236841, 7.4904494783373894031687405107, 7.76702480520236574429746779494, 8.66877799435894904409838168827, 9.855899666121149094748291402333, 10.40078710307363145596225674584, 11.42243092917656902761014505250, 12.44268178187612880024696768327, 13.178707306970862082791350690836, 14.03300998639617974385137864935, 14.86682601065406699427490730358, 15.60591534182497888272741055515, 15.87636872067537874347549998695, 17.47794022121288727950667480651, 18.1469194457872180513351142623, 19.01505202032046226628914454513, 19.52925585938418792445025997507, 20.35136281800380831089791132353, 21.05399730906875860093734948441, 21.86368697977507241853057341830

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