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Properties

Label 1-1096-1096.1003-r1-0-0
Degree $1$
Conductor $1096$
Sign $-0.999 + 0.0109i$
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  + (0.961 + 0.273i)3-s + (0.361 − 0.932i)5-s + (−0.602 + 0.798i)7-s + (0.850 + 0.526i)9-s + (−0.739 + 0.673i)11-s + (−0.798 − 0.602i)13-s + (0.602 − 0.798i)15-s + (−0.445 − 0.895i)17-s + (0.982 + 0.183i)19-s + (−0.798 + 0.602i)21-s + (0.995 + 0.0922i)23-s + (−0.739 − 0.673i)25-s + (0.673 + 0.739i)27-s + (−0.995 − 0.0922i)29-s + (−0.183 − 0.982i)31-s + ⋯
L(s)  = 1  + (0.961 + 0.273i)3-s + (0.361 − 0.932i)5-s + (−0.602 + 0.798i)7-s + (0.850 + 0.526i)9-s + (−0.739 + 0.673i)11-s + (−0.798 − 0.602i)13-s + (0.602 − 0.798i)15-s + (−0.445 − 0.895i)17-s + (0.982 + 0.183i)19-s + (−0.798 + 0.602i)21-s + (0.995 + 0.0922i)23-s + (−0.739 − 0.673i)25-s + (0.673 + 0.739i)27-s + (−0.995 − 0.0922i)29-s + (−0.183 − 0.982i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $-0.999 + 0.0109i$
Analytic conductor: \(117.781\)
Root analytic conductor: \(117.781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1096,\ (1:\ ),\ -0.999 + 0.0109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0005357462777 + 0.09774877526i\)
\(L(\frac12)\) \(\approx\) \(0.0005357462777 + 0.09774877526i\)
\(L(1)\) \(\approx\) \(1.156794012 + 0.04117050373i\)
\(L(1)\) \(\approx\) \(1.156794012 + 0.04117050373i\)

Euler product

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

−20.802030683899086862961368759316, −19.91048156321489168057107746559, −19.270877700423413107166285506724, −18.71401425428205386664817082124, −17.91176707097229206447158141212, −16.97357174057548390375660569553, −16.08850673445005848564749112697, −15.14057671993302335334535186910, −14.543415222115592813791371270828, −13.62337549652900839508756715688, −13.35260984218923014026015556820, −12.3487537117143794881489359805, −11.081801176230577451256101885035, −10.415049089936490625174594315316, −9.63115437807985048838164860379, −8.885607097549804864213535096586, −7.72263926769702433965899879131, −7.11903632356257822737146661985, −6.500888974169141373857372403149, −5.29362409543287837866456306611, −3.96251925072071347392173662317, −3.22760636714296511209568407712, −2.52162254555636094961951104586, −1.45457274807234535574635450491, −0.015429335830530569909024771224, 1.447581605055403505829750426731, 2.57647104759828633047090629189, 3.01958010316888748935660253877, 4.52191769124237735174747439230, 5.04926181367796067954816113151, 5.9981498838050072754816032570, 7.43871597216386037377928933667, 7.87582587443878636905360590749, 9.11936807351851652390303501737, 9.45181042810080083736532771315, 10.04325831240892359764860041693, 11.38378623080466207610742247474, 12.489080904201315690073352978523, 13.00875422115875405716953579174, 13.56609145034159387492794625211, 14.7868350225306135030189938124, 15.30631675821105138345348774523, 16.07919297176450474365489415185, 16.736653112802037304746647713657, 17.92991595142068882373694109409, 18.5021609014743987695345397295, 19.49111504759124945177504049339, 20.24719609738736860709957873893, 20.627652145712854199487103211780, 21.49948736683987609812310338970

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