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Properties

Label 1-896-896.165-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: $\chi_{896} (165, \cdot )$
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

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

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