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Properties

Label 1-896-896.51-r1-0-0
Degree $1$
Conductor $896$
Sign $0.801 - 0.597i$
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  + (−0.0654 − 0.997i)3-s + (−0.659 − 0.751i)5-s + (−0.991 + 0.130i)9-s + (0.442 + 0.896i)11-s + (−0.980 − 0.195i)13-s + (−0.707 + 0.707i)15-s + (−0.965 + 0.258i)17-s + (−0.946 + 0.321i)19-s + (−0.130 − 0.991i)23-s + (−0.130 + 0.991i)25-s + (0.195 + 0.980i)27-s + (0.555 + 0.831i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s + (0.751 − 0.659i)37-s + ⋯
L(s)  = 1  + (−0.0654 − 0.997i)3-s + (−0.659 − 0.751i)5-s + (−0.991 + 0.130i)9-s + (0.442 + 0.896i)11-s + (−0.980 − 0.195i)13-s + (−0.707 + 0.707i)15-s + (−0.965 + 0.258i)17-s + (−0.946 + 0.321i)19-s + (−0.130 − 0.991i)23-s + (−0.130 + 0.991i)25-s + (0.195 + 0.980i)27-s + (0.555 + 0.831i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s + (0.751 − 0.659i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.801 - 0.597i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ 0.801 - 0.597i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8637526782 - 0.2863394570i\)
\(L(\frac12)\) \(\approx\) \(0.8637526782 - 0.2863394570i\)
\(L(1)\) \(\approx\) \(0.7025391717 - 0.2751702183i\)
\(L(1)\) \(\approx\) \(0.7025391717 - 0.2751702183i\)

Euler product

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

−21.81244493398844707114551290763, −21.38179126707534194850077652690, −20.052586301008010573205053429094, −19.60874511917424294345870622955, −18.87505416266353851811265379049, −17.67251108171835211185010708189, −17.06044337794007689869389462788, −16.04776175862012894646543240239, −15.53980449005619197531713592136, −14.64816049224540190046072512823, −14.17587909245545618922125296494, −13.00687231886592590642120455210, −11.64533083844140896074781250145, −11.374100170764284283463015642257, −10.50614679582921156447453420823, −9.62306909426569369789216869243, −8.78840933577222523008451508906, −7.90053186529394894775329084095, −6.7994322241166651256203474908, −6.012838151278841198496559491027, −4.80278964193385600073237938636, −4.076404259748137621267343992318, −3.19636193446112540819737461267, −2.319429697424287783471580998300, −0.36663959908119301983663033137, 0.52523837355241577020708980417, 1.74707994846313867033740643063, 2.54819080431676435474969187764, 4.05288183322684956950843667072, 4.764902758640330146831437967343, 5.88376380348951799489072968102, 6.92999064940783366705654004630, 7.49931394831713339860973434330, 8.49880457612425414131935016787, 9.070821889885777292904944836622, 10.34895632329998757269169493896, 11.35449171188900175260867879988, 12.27665966548575236389105088483, 12.59326205816379819376570156495, 13.36389227387527086811409975130, 14.66339441418308284251722884678, 14.995877739382233697241893315457, 16.34009930498253771289436335874, 16.96892287569197211110571099371, 17.6973441881884880297604799203, 18.50136184722505966285479342818, 19.56255112776287722303410969076, 19.872850478742150006618177479935, 20.54831613389922650605618445022, 21.82249762964352304154538981324

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