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Properties

Label 1-1096-1096.117-r1-0-0
Degree $1$
Conductor $1096$
Sign $0.813 + 0.582i$
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  + (−0.317 − 0.948i)3-s + (−0.769 + 0.638i)5-s + (−0.526 + 0.850i)7-s + (−0.798 + 0.602i)9-s + (−0.183 − 0.982i)11-s + (−0.973 + 0.228i)13-s + (0.850 + 0.526i)15-s + (−0.961 − 0.273i)17-s + (−0.673 − 0.739i)19-s + (0.973 + 0.228i)21-s + (0.403 − 0.914i)23-s + (0.183 − 0.982i)25-s + (0.824 + 0.565i)27-s + (−0.914 − 0.403i)29-s + (−0.998 + 0.0461i)31-s + ⋯
L(s)  = 1  + (−0.317 − 0.948i)3-s + (−0.769 + 0.638i)5-s + (−0.526 + 0.850i)7-s + (−0.798 + 0.602i)9-s + (−0.183 − 0.982i)11-s + (−0.973 + 0.228i)13-s + (0.850 + 0.526i)15-s + (−0.961 − 0.273i)17-s + (−0.673 − 0.739i)19-s + (0.973 + 0.228i)21-s + (0.403 − 0.914i)23-s + (0.183 − 0.982i)25-s + (0.824 + 0.565i)27-s + (−0.914 − 0.403i)29-s + (−0.998 + 0.0461i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.813 + 0.582i$
Analytic conductor: \(117.781\)
Root analytic conductor: \(117.781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1096,\ (1:\ ),\ 0.813 + 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05458865195 + 0.01752489027i\)
\(L(\frac12)\) \(\approx\) \(0.05458865195 + 0.01752489027i\)
\(L(1)\) \(\approx\) \(0.4965863721 - 0.1486285315i\)
\(L(1)\) \(\approx\) \(0.4965863721 - 0.1486285315i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
137 \( 1 \)
good3 \( 1 + (-0.317 - 0.948i)T \)
5 \( 1 + (-0.769 + 0.638i)T \)
7 \( 1 + (-0.526 + 0.850i)T \)
11 \( 1 + (-0.183 - 0.982i)T \)
13 \( 1 + (-0.973 + 0.228i)T \)
17 \( 1 + (-0.961 - 0.273i)T \)
19 \( 1 + (-0.673 - 0.739i)T \)
23 \( 1 + (0.403 - 0.914i)T \)
29 \( 1 + (-0.914 - 0.403i)T \)
31 \( 1 + (-0.998 + 0.0461i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.0461 - 0.998i)T \)
47 \( 1 + (0.138 - 0.990i)T \)
53 \( 1 + (-0.998 - 0.0461i)T \)
59 \( 1 + (0.602 + 0.798i)T \)
61 \( 1 + (0.798 + 0.602i)T \)
67 \( 1 + (0.228 + 0.973i)T \)
71 \( 1 + (0.824 - 0.565i)T \)
73 \( 1 + (-0.850 + 0.526i)T \)
79 \( 1 + (-0.948 - 0.317i)T \)
83 \( 1 + (-0.873 - 0.486i)T \)
89 \( 1 + (-0.638 - 0.769i)T \)
97 \( 1 + (-0.565 + 0.824i)T \)
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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

−21.00568578437207613253335108049, −20.230094326315843979183000304382, −19.957568966212024746769410929914, −19.06440317804468200917477985832, −17.72457496254306554054245624997, −17.028328993925821027281498115473, −16.59641411042620349085700674201, −15.63631392576204327134724384721, −15.128656585627799758825454020324, −14.349573432980317858220024349006, −12.913681744586400330057286539664, −12.65118252351980723687387138349, −11.470794714515218139597074703834, −10.86659085451908253352193779012, −9.82632702631240041076463348511, −9.4534576918856994095766601742, −8.29321719524344966309256451829, −7.4324350170710781761123579225, −6.57086263097500615774384215670, −5.31219821346841296944724968558, −4.58113567953960287496117420896, −3.97406800597178940907239319557, −3.0854135697362717328142442322, −1.56302874285513065342337314834, −0.03472043182770977661818697929, 0.327177095971665752381017709650, 2.23784278409176098265237219579, 2.6288220423901427853533499915, 3.8046980657369761265244691436, 5.08777929134910322120860569980, 5.956797599033979953277472054965, 6.872664871516129420147797785781, 7.28219591516068336118438818874, 8.52133278888743325184112710501, 8.9506320310347701176982747552, 10.41670862908606687475033623740, 11.23258467800094423402534630135, 11.74902753113259100589013396516, 12.651487047330545074257763887183, 13.22782549856098472462624955149, 14.29560166296160392919165759129, 15.01294072756850064860126408395, 15.86366578914246178061668909244, 16.65475193039395850702046389495, 17.53348501057238266404754658086, 18.5109998266526150727286058752, 18.89071980124198828237393729527, 19.48200050633903821035412650803, 20.21385024898936366537550451966, 21.64938083655792077468067079869

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