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Properties

Label 1-896-896.341-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.650 + 0.759i$
Analytic cond. $96.2885$
Root an. cond. $96.2885$
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  + (−0.946 − 0.321i)3-s + (0.896 + 0.442i)5-s + (0.793 + 0.608i)9-s + (−0.659 + 0.751i)11-s + (0.831 + 0.555i)13-s + (−0.707 − 0.707i)15-s + (0.258 + 0.965i)17-s + (0.997 + 0.0654i)19-s + (−0.608 + 0.793i)23-s + (0.608 + 0.793i)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 + (−0.442 + 0.896i)37-s + ⋯
L(s)  = 1  + (−0.946 − 0.321i)3-s + (0.896 + 0.442i)5-s + (0.793 + 0.608i)9-s + (−0.659 + 0.751i)11-s + (0.831 + 0.555i)13-s + (−0.707 − 0.707i)15-s + (0.258 + 0.965i)17-s + (0.997 + 0.0654i)19-s + (−0.608 + 0.793i)23-s + (0.608 + 0.793i)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 + (−0.442 + 0.896i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.650 + 0.759i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.650 + 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5853639777 + 1.271218128i\)
\(L(\frac12)\) \(\approx\) \(0.5853639777 + 1.271218128i\)
\(L(1)\) \(\approx\) \(0.9014540807 + 0.2471064824i\)
\(L(1)\) \(\approx\) \(0.9014540807 + 0.2471064824i\)

Euler product

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

−21.49952415479836328865419768359, −20.71879544536713460394847258554, −20.319072954578307236240393334242, −18.57633946461599625367907428690, −18.363600560831357386763092228577, −17.476610041864430191725249194312, −16.627843327808114630163953002054, −16.07973650584525345353144609505, −15.365549636888146762309073351674, −13.98835326943368850199057831592, −13.44193240939202310380937041113, −12.56250961468967761696939600522, −11.68922462302532689367882067933, −10.82710688094379709283488789594, −10.11486329002496529937254963858, −9.35530116272366878250066301931, −8.379899432633203907907181356617, −7.26294481670732130671067365422, −6.094661442135945814838164341587, −5.605937168346381147598712444043, −4.89855694063505262128670672111, −3.70602212326435326996598917765, −2.54644090043937108278968205645, −1.14349813418221369985303706112, −0.3759056532354179928222666265, 1.33616286881022865647288872016, 1.92141578712041287811101732433, 3.293737353110864062678605547, 4.53717303720631116628717238423, 5.55316038868863331166823739928, 6.075782630857658816757842385414, 7.02242935841087510387269077143, 7.76432139677702519366351791988, 9.07570113337476069179194206627, 10.12843251607491144901230274842, 10.5049646122360246289045296104, 11.5503790901206704257946512642, 12.27868944741664496428784539616, 13.33858576775363920434662173173, 13.68656053166554096011457950291, 14.90656336127493868753313818786, 15.76682462036644353310089471122, 16.61837477285117495347012064433, 17.43599888562895443213708099127, 18.050102304587063644062802092792, 18.57564228825001215151875101530, 19.48283904236236403917624145358, 20.77975240782472610223109191086, 21.267594749260954517866886055634, 22.2065052609068035469504597214

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