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Properties

Label 1-1096-1096.339-r1-0-0
Degree $1$
Conductor $1096$
Sign $0.999 + 0.0208i$
Analytic cond. $117.781$
Root an. cond. $117.781$
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  + (−0.0922 − 0.995i)3-s + (−0.602 − 0.798i)5-s + (−0.739 + 0.673i)7-s + (−0.982 + 0.183i)9-s + (−0.273 − 0.961i)11-s + (0.739 − 0.673i)13-s + (−0.739 + 0.673i)15-s + (0.932 − 0.361i)17-s + (0.445 − 0.895i)19-s + (0.739 + 0.673i)21-s + (−0.850 + 0.526i)23-s + (−0.273 + 0.961i)25-s + (0.273 + 0.961i)27-s + (−0.850 + 0.526i)29-s + (0.445 + 0.895i)31-s + ⋯
L(s)  = 1  + (−0.0922 − 0.995i)3-s + (−0.602 − 0.798i)5-s + (−0.739 + 0.673i)7-s + (−0.982 + 0.183i)9-s + (−0.273 − 0.961i)11-s + (0.739 − 0.673i)13-s + (−0.739 + 0.673i)15-s + (0.932 − 0.361i)17-s + (0.445 − 0.895i)19-s + (0.739 + 0.673i)21-s + (−0.850 + 0.526i)23-s + (−0.273 + 0.961i)25-s + (0.273 + 0.961i)27-s + (−0.850 + 0.526i)29-s + (0.445 + 0.895i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.999 + 0.0208i$
Analytic conductor: \(117.781\)
Root analytic conductor: \(117.781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1096,\ (1:\ ),\ 0.999 + 0.0208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6966408561 + 0.007254785070i\)
\(L(\frac12)\) \(\approx\) \(0.6966408561 + 0.007254785070i\)
\(L(1)\) \(\approx\) \(0.6710589132 - 0.3175975185i\)
\(L(1)\) \(\approx\) \(0.6710589132 - 0.3175975185i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
137 \( 1 \)
good3 \( 1 + (-0.0922 - 0.995i)T \)
5 \( 1 + (-0.602 - 0.798i)T \)
7 \( 1 + (-0.739 + 0.673i)T \)
11 \( 1 + (-0.273 - 0.961i)T \)
13 \( 1 + (0.739 - 0.673i)T \)
17 \( 1 + (0.932 - 0.361i)T \)
19 \( 1 + (0.445 - 0.895i)T \)
23 \( 1 + (-0.850 + 0.526i)T \)
29 \( 1 + (-0.850 + 0.526i)T \)
31 \( 1 + (0.445 + 0.895i)T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + (-0.445 + 0.895i)T \)
47 \( 1 + (-0.982 + 0.183i)T \)
53 \( 1 + (0.445 - 0.895i)T \)
59 \( 1 + (-0.982 + 0.183i)T \)
61 \( 1 + (0.982 + 0.183i)T \)
67 \( 1 + (-0.739 + 0.673i)T \)
71 \( 1 + (-0.273 + 0.961i)T \)
73 \( 1 + (0.739 + 0.673i)T \)
79 \( 1 + (0.0922 - 0.995i)T \)
83 \( 1 + (-0.932 - 0.361i)T \)
89 \( 1 + (0.602 + 0.798i)T \)
97 \( 1 + (0.273 + 0.961i)T \)
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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

−21.09917538132027329009240664587, −20.52216100793205208111529223279, −19.821824857408229733978284990920, −18.87831072043666217670652628039, −18.2857419530562715610770966607, −17.04259717201206634945256037553, −16.53692096540834057357579117854, −15.71052082514341234411316718610, −15.12856731968658383436820042249, −14.27867115052573665314054988929, −13.620332317748382471853801593835, −12.295617480362662433737846953290, −11.69925010849070308013813467019, −10.65997778398284697451581367885, −10.132569446788818416075965363742, −9.57480092953139295646222141504, −8.30757013953024456659708683877, −7.52433815633417394562909787630, −6.5485095603087393198144926875, −5.78810594440907233904054573432, −4.5065780412153305544223770271, −3.75578354974562501638482256679, −3.3014898427302565225694666423, −1.9404804724639244960718744510, −0.22229224795786104979334408306, 0.65616794603283344267710450787, 1.56601624006771358223320322333, 3.04454694230716876473076304154, 3.44635486595421247135575685126, 5.20810907266183425189911313510, 5.61495061348426234039542706766, 6.60835561549698951734747102490, 7.58644045474822095270660037, 8.39906658028771469841289366694, 8.861003433444799235292023008139, 10.00463516182912958839444515189, 11.28446962405314443092318250379, 11.79953149299605390379134305340, 12.6162792310222088210349939281, 13.23180270057431207458103376334, 13.85320237665037946224712630378, 15.06492807723191655602637200608, 16.10480801024163694927522883923, 16.264979625175874534991458795743, 17.46305415343108451413015351764, 18.27149933356900162014420316809, 18.92502810082604576088305237997, 19.586372366843695371237887423438, 20.22585859444405836719953955818, 21.153607657581388480688269382382

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