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Properties

Label 1-896-896.395-r0-0-0
Degree $1$
Conductor $896$
Sign $-0.572 - 0.819i$
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  + (−0.659 − 0.751i)3-s + (−0.997 + 0.0654i)5-s + (−0.130 + 0.991i)9-s + (0.946 − 0.321i)11-s + (−0.555 + 0.831i)13-s + (0.707 + 0.707i)15-s + (−0.965 − 0.258i)17-s + (−0.896 + 0.442i)19-s + (0.991 + 0.130i)23-s + (0.991 − 0.130i)25-s + (0.831 − 0.555i)27-s + (−0.195 − 0.980i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s + (−0.0654 − 0.997i)37-s + ⋯
L(s)  = 1  + (−0.659 − 0.751i)3-s + (−0.997 + 0.0654i)5-s + (−0.130 + 0.991i)9-s + (0.946 − 0.321i)11-s + (−0.555 + 0.831i)13-s + (0.707 + 0.707i)15-s + (−0.965 − 0.258i)17-s + (−0.896 + 0.442i)19-s + (0.991 + 0.130i)23-s + (0.991 − 0.130i)25-s + (0.831 − 0.555i)27-s + (−0.195 − 0.980i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s + (−0.0654 − 0.997i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.572 - 0.819i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.572 - 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2500866168 - 0.4796476103i\)
\(L(\frac12)\) \(\approx\) \(0.2500866168 - 0.4796476103i\)
\(L(1)\) \(\approx\) \(0.6273614680 - 0.1810107723i\)
\(L(1)\) \(\approx\) \(0.6273614680 - 0.1810107723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.659 - 0.751i)T \)
5 \( 1 + (-0.997 + 0.0654i)T \)
11 \( 1 + (0.946 - 0.321i)T \)
13 \( 1 + (-0.555 + 0.831i)T \)
17 \( 1 + (-0.965 - 0.258i)T \)
19 \( 1 + (-0.896 + 0.442i)T \)
23 \( 1 + (0.991 + 0.130i)T \)
29 \( 1 + (-0.195 - 0.980i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (-0.0654 - 0.997i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (-0.980 - 0.195i)T \)
47 \( 1 + (0.258 + 0.965i)T \)
53 \( 1 + (0.946 - 0.321i)T \)
59 \( 1 + (0.442 - 0.896i)T \)
61 \( 1 + (-0.321 + 0.946i)T \)
67 \( 1 + (-0.659 - 0.751i)T \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (-0.793 - 0.608i)T \)
79 \( 1 + (0.965 - 0.258i)T \)
83 \( 1 + (-0.831 - 0.555i)T \)
89 \( 1 + (-0.608 - 0.793i)T \)
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

−22.29438704781660387254665394529, −21.640408613338173240549565126774, −20.553579471843536832426224623940, −19.91036638202484743811954715925, −19.27057259280111217022398531440, −18.105660924888850196102116614463, −17.233139706163009852146990839175, −16.78936061444906266979042942813, −15.731001262675814097001206656893, −15.11322079234008456957427785085, −14.68889916275180563662977174354, −13.17712715588025953507538046837, −12.31622593401156241527534565473, −11.69372054296848534960545145043, −10.848589125251424745176965745446, −10.22872843072112686834170404571, −9.01449756534080326320422673878, −8.50695747123907627664049252962, −7.0869150798521120254416074018, −6.58414243637624439813740328629, −5.250989091297279423685107847628, −4.53315677603806433045317314824, −3.80670353355629184224461783751, −2.77728024507255921602003403065, −1.04237048202365841325041054379, 0.31991758505224339475219916885, 1.615297489073591796778617609176, 2.72466818626719415870497977863, 4.12723568922631525807787072649, 4.67892783718800807021182140922, 6.06004302559548709125057931944, 6.74699131299909487128949993113, 7.45062630756829657225264247745, 8.40070398087991017269157574718, 9.23776253505151555597195850375, 10.56046002768984014009649382036, 11.49680330895032501455276711795, 11.72032423517587208571347622112, 12.69221358017249955256444076861, 13.50470390622946565205373218191, 14.49432577288139924877786924065, 15.265719297117140274686468370123, 16.354353222633457199273795947308, 16.88189239869426812978102328652, 17.64544600937620600876309932682, 18.700608706124294189308026920921, 19.362592198794926391161893678132, 19.631386562691955325272591550720, 20.92472847066310070159859916849, 21.89285287204218869185251726838

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