Properties

Degree $1$
Conductor $88$
Sign $0.569 - 0.821i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.569 - 0.821i$
Motivic weight: \(0\)
Character: $\chi_{88} (35, \cdot )$
Sato-Tate group: $\mu(10)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 88,\ (0:\ ),\ 0.569 - 0.821i)\)

Particular Values

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

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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