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Properties

Label 1-88-88.83-r0-0-0
Degree $1$
Conductor $88$
Sign $0.569 + 0.821i$
Analytic cond. $0.408670$
Root an. cond. $0.408670$
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  + (0.309 + 0.951i)3-s + (0.809 + 0.587i)5-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + 21-s − 23-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.309 − 0.951i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)35-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)3-s + (0.809 + 0.587i)5-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + 21-s − 23-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.309 − 0.951i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(0.408670\)
Root analytic conductor: \(0.408670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 88,\ (0:\ ),\ 0.569 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.040449367 + 0.5446240330i\)
\(L(\frac12)\) \(\approx\) \(1.040449367 + 0.5446240330i\)
\(L(1)\) \(\approx\) \(1.142383617 + 0.3798421513i\)
\(L(1)\) \(\approx\) \(1.142383617 + 0.3798421513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 - T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−30.25390850927634962796753323886, −29.44695491677101531649213707265, −28.52698204617063602606724347533, −27.4220593598833949402175924090, −25.787339131647005963499416353234, −24.95842375285006053815589974076, −24.47575762229724728676082904519, −23.12754505561220611730017335663, −21.776785178899816512762581918838, −20.743212642920019717948744803473, −19.64747933974470240958317464162, −18.378725201264570783108778371, −17.711366514728755144087432343732, −16.4421781025370931337430124864, −14.7852044221369024832801760500, −13.90577643877920234209840303841, −12.557918627012724239751069994697, −12.03084489259707800525121997619, −10.02346327607988422780582070306, −8.79442984100861191828717083010, −7.790071487837242183990935469817, −6.16277078238344649977709960549, −5.18089814602901809182168957544, −2.80844991148384309018647478078, −1.578492752311248983020723648960, 2.26154930418065263055477996661, 3.80569895171860997269491126206, 5.08878051463387761901661863336, 6.6563042597998997521755774833, 8.14999257936768227040307830559, 9.75243069277853918667019499034, 10.29770193062558372518264118444, 11.562828327765076894052408067430, 13.54287570861749282327951220562, 14.28899502422135237897441161678, 15.28032461439154754565399643344, 16.828016874652175582936700593233, 17.40304777249728851897203024299, 19.05525198241342034126262154093, 20.18379408985967980126400175692, 21.284056593179810807835895775354, 21.962285294724529872772340861125, 23.14401005624454322586018125549, 24.48199546566299474780111354892, 25.9593803293387472697544019108, 26.31658953354176563622921817389, 27.43296684217256784057980966342, 28.5851664444849394376830357695, 29.809458642062843065954192621441, 30.622709773045772003894044022749

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