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Properties

Label 1-896-896.429-r0-0-0
Degree $1$
Conductor $896$
Sign $0.801 - 0.597i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.801 - 0.597i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.801 - 0.597i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.443990820 - 0.8101983612i\)
\(L(\frac12)\) \(\approx\) \(2.443990820 - 0.8101983612i\)
\(L(1)\) \(\approx\) \(1.705802841 - 0.2918809765i\)
\(L(1)\) \(\approx\) \(1.705802841 - 0.2918809765i\)

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

−22.01963958808785025388457326192, −21.22430413191550324729762247898, −20.3792648231275110806450040260, −19.73424170374999797106422183168, −18.98595610184681636481903923486, −17.9969507141517849569003388200, −17.5428315841149618492649895619, −16.42044407879680635937790482766, −15.25602246956389110689409326144, −14.90551831124297916367686853861, −14.00185658703208623065818453875, −13.41312201309663767627484272314, −12.54785169664206172712117393592, −11.46164144572606647082742919419, −10.2767622378736774507615843466, −9.873709388166800266354545510003, −8.964655080756464798381011279182, −8.08655278244356778495170134931, −7.10393251954224204588470248571, −6.41666875141432291541796330177, −5.260224812656148855694775476984, −4.08917823765337251149846368459, −3.217899461217511833091723368532, −2.30136842635625195647765807048, −1.46847509512309676852919066831, 1.12729819293072996547613786585, 2.06634427090997126028436794298, 2.91167471276318016263007346174, 4.321788959594130464539087016529, 4.68381349440274377108417161892, 6.43402696570381063639670891879, 6.64839646042062981364503464570, 8.20276866388609444676066014508, 8.77338615364890777873134603442, 9.39148917311151136605494259831, 10.19978348982700106900220348291, 11.39530344689733622255010158287, 12.35751392407853045533035106355, 13.22807649131231349182152608322, 13.848580643020407785360829990499, 14.4529734132716263953033455152, 15.43405851614775346901219113927, 16.30312865650191388588955501542, 17.11861301261221656388865008509, 17.87190464506302311538143183953, 18.920383462106983774985965634131, 19.60277514899734096863355686942, 20.21467244994118318199726303363, 21.13213724397464778626467883309, 21.6697104491558065902372968887

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