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Properties

Label 1-896-896.579-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  + (0.751 + 0.659i)3-s + (0.0654 − 0.997i)5-s + (0.130 + 0.991i)9-s + (−0.321 + 0.946i)11-s + (−0.831 + 0.555i)13-s + (0.707 − 0.707i)15-s + (−0.965 + 0.258i)17-s + (−0.442 + 0.896i)19-s + (−0.991 + 0.130i)23-s + (−0.991 − 0.130i)25-s + (−0.555 + 0.831i)27-s + (0.980 + 0.195i)29-s + (0.866 + 0.5i)31-s + (−0.866 + 0.5i)33-s + (−0.997 − 0.0654i)37-s + ⋯
L(s)  = 1  + (0.751 + 0.659i)3-s + (0.0654 − 0.997i)5-s + (0.130 + 0.991i)9-s + (−0.321 + 0.946i)11-s + (−0.831 + 0.555i)13-s + (0.707 − 0.707i)15-s + (−0.965 + 0.258i)17-s + (−0.442 + 0.896i)19-s + (−0.991 + 0.130i)23-s + (−0.991 − 0.130i)25-s + (−0.555 + 0.831i)27-s + (0.980 + 0.195i)29-s + (0.866 + 0.5i)31-s + (−0.866 + 0.5i)33-s + (−0.997 − 0.0654i)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: $\chi_{896} (579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.572 + 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5366765137 + 1.029305808i\)
\(L(\frac12)\) \(\approx\) \(0.5366765137 + 1.029305808i\)
\(L(1)\) \(\approx\) \(1.035657496 + 0.3525796949i\)
\(L(1)\) \(\approx\) \(1.035657496 + 0.3525796949i\)

Euler product

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

−21.733484416628331357912337508761, −20.87001817854634658258294807976, −19.78734178070846392581530512429, −19.39406680625163873683889517763, −18.58314190096550036685477421675, −17.83599735524408584629776337548, −17.2514762470699200990377659143, −15.65971384523742691205821403613, −15.369866139141280229176602198244, −14.198455248873962492797080586716, −13.84606112504096997421064604096, −12.971559821643480611790790401828, −12.013684927156170016008109419246, −11.11565572907291935872216672933, −10.2581150247227232679079346364, −9.34455327496147631828098186293, −8.32091509427374345985814247400, −7.70049065524594621329525011723, −6.6838451323670713848492115666, −6.1937656146404293121663374525, −4.78938161987286275796592908333, −3.53043248763983946921065140983, −2.71656200267474887513180433974, −2.1221214031844505261404085144, −0.41812275953487752720278734405, 1.72107473683879870877409811217, 2.36086865746983478566957110670, 3.779287749655424630449179496206, 4.57776845124759816829579198896, 5.087404640287011050551089464626, 6.45434710781964531019651492933, 7.62125356046931873271502002491, 8.38646724617595729260827786872, 9.10265646539026042251343361261, 9.954633933008356073752962653603, 10.49208298326789996943719286166, 11.9790411158332538793421129546, 12.47440826533850695937930715246, 13.562624091966931884395485169732, 14.14346657845497443210379662647, 15.23413828961478881847413712994, 15.68351206912931884249223707971, 16.65477851516701126708821088699, 17.252145735595028003173161951808, 18.262284744337026503389914571584, 19.46513754947854245134949616263, 19.91162239130808873829386339744, 20.60533873628303779160633168302, 21.36848630780829742774029646441, 21.94127447914828030789613648982

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