Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $0.998 + 0.0500i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.415 + 0.909i)2-s + (0.654 − 0.755i)3-s + (−0.654 − 0.755i)4-s + (0.959 − 0.281i)5-s + (0.415 + 0.909i)6-s + (−0.959 + 0.281i)7-s + (0.959 − 0.281i)8-s + (−0.142 − 0.989i)9-s + (−0.142 + 0.989i)10-s + (0.959 + 0.281i)11-s − 12-s + (0.142 − 0.989i)14-s + (0.415 − 0.909i)15-s + (−0.142 + 0.989i)16-s + (0.415 + 0.909i)17-s + (0.959 + 0.281i)18-s + ⋯
L(s,χ)  = 1  + (−0.415 + 0.909i)2-s + (0.654 − 0.755i)3-s + (−0.654 − 0.755i)4-s + (0.959 − 0.281i)5-s + (0.415 + 0.909i)6-s + (−0.959 + 0.281i)7-s + (0.959 − 0.281i)8-s + (−0.142 − 0.989i)9-s + (−0.142 + 0.989i)10-s + (0.959 + 0.281i)11-s − 12-s + (0.142 − 0.989i)14-s + (0.415 − 0.909i)15-s + (−0.142 + 0.989i)16-s + (0.415 + 0.909i)17-s + (0.959 + 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.998 + 0.0500i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.998 + 0.0500i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $0.998 + 0.0500i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (441, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 1157,\ (0:\ ),\ 0.998 + 0.0500i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.715942472 + 0.04296098593i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.715942472 + 0.04296098593i\)
\(L(\chi,1)\)  \(\approx\)  \(1.186087589 + 0.1062876823i\)
\(L(1,\chi)\)  \(\approx\)  \(1.186087589 + 0.1062876823i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.25332353295805941935943377780, −20.34803345598292810656649812954, −20.00523656051412947898956211466, −18.871636472780230263457049548689, −18.635339359112868056940443110871, −17.35385527929500180279666875266, −16.614494724337827629514298536763, −16.2533356163236196982584099680, −14.77887827112916128444007307055, −14.10450240174375895947343772588, −13.538775130128817245851073977305, −12.70152757277430915995482127892, −11.72598528600695369198616484805, −10.662809732159198306733863613274, −10.11996220502530246015645641633, −9.46834013121534818298351146109, −8.98401353090961785479182943634, −7.9607095246650423458605214195, −6.84777366227866027116019284229, −5.82167312590481411323479982962, −4.61280865708772439139344067081, −3.74937754324940707254532203840, −2.97445687464949982905939699153, −2.289708017955733150863280853317, −1.04073631716616317744844130181, 0.96173974224157596254877232905, 1.78848386221090735025206550159, 2.95160867199019257195581676209, 4.11915518944830689294247290514, 5.37272960323456444786202512283, 6.37134129240065376687972304949, 6.55617254598960390026747283793, 7.618605961950352414157400096170, 8.603734671268195926644135528105, 9.344700183518085184635036871571, 9.590751869361040810808376849338, 10.76853165441557754990151069964, 12.32178764759602672274092554397, 12.81181506963171582218254865450, 13.76673585342017432449357200633, 14.120167641332322531698912344137, 15.133746011553516935157145110699, 15.75957820480681206088245203945, 16.94553764064151704755663904255, 17.35695649670456382510051221225, 18.07691815273774129875593096339, 18.9825406674612788552508027099, 19.57381993654291050869781853935, 20.09716128471680071636321764447, 21.454465609749194251598231994236

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