Properties

Degree 1
Conductor $ 2^{3} \cdot 7 $
Sign $0.386 - 0.922i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + 13-s + 15-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s − 29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + ⋯
L(s,χ)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + 13-s + 15-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s − 29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.386 - 0.922i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.386 - 0.922i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(56\)    =    \(2^{3} \cdot 7\)
\( \varepsilon \)  =  $0.386 - 0.922i$
motivic weight  =  \(0\)
character  :  $\chi_{56} (5, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 56,\ (1:\ ),\ 0.386 - 0.922i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8207880187 - 0.5459733256i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8207880187 - 0.5459733256i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8328904045 - 0.1061381970i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8328904045 - 0.1061381970i\)

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

−33.303190839132080343342023080206, −31.54426221373143704525488251646, −30.396777748075247186238429915092, −29.97096474977585039755147784180, −28.42051650161422327708627593285, −27.56459954820951227424890161982, −25.978032504612118492662790419023, −25.092500195710263402839434735537, −23.47629534286427331197175326959, −23.05956352751087535748522071738, −21.75718188401715853439926326543, −19.98578246234125872338731830738, −18.90313038073641507621087051328, −18.0279745963835417083146788612, −16.79708786168985219741984886888, −15.23698815476838499339514135274, −14.003169338092966937197428270498, −12.55658441379179326952439002714, −11.50197194818431498963583880353, −10.31464188653018778364807097620, −8.20704584444199725366806153043, −7.028480989586577041650385685318, −5.898906527966484116955254070158, −3.76052958575870632124191886021, −1.73250857361019227909332214378, 0.61143003446968288555295849038, 3.58091852871631669829789861368, 4.8293321138044502499171325478, 6.22115016599222491301287011002, 8.354611778351419843284759161523, 9.40368565478028008471827981517, 11.03368982274204631596498235210, 11.93023590152752959682407220236, 13.52409807795634397353438546019, 15.17237094910416958100298558312, 16.267694875051908751783799453753, 16.94989854042810753062585736632, 18.59244678460921910002699420201, 20.14890543190247601488888469434, 21.000356189615498531901517442716, 22.22008816712626582141357215530, 23.38064085557850779850195030591, 24.400984912708843331777041346336, 25.92777454988139201417122187758, 27.20106867896025280707143296272, 27.93744762664238255381557106643, 28.87334563532176514710159460243, 30.27636903208698263528280715912, 31.89571297118287268128528324895, 32.36506566043854788882292500724

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