Properties

Degree 1
Conductor $ 7 \cdot 11 $
Sign $0.970 - 0.242i$
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)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s − 12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (0.809 + 0.587i)18-s + (−0.809 + 0.587i)19-s + (−0.309 + 0.951i)20-s + ⋯
L(s,χ)  = 1  + (−0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s − 12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (0.809 + 0.587i)18-s + (−0.809 + 0.587i)19-s + (−0.309 + 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.970 - 0.242i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.970 - 0.242i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(77\)    =    \(7 \cdot 11\)
\( \varepsilon \)  =  $0.970 - 0.242i$
motivic weight  =  \(0\)
character  :  $\chi_{77} (41, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 77,\ (0:\ ),\ 0.970 - 0.242i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9253962889 - 0.1137557129i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9253962889 - 0.1137557129i\)
\(L(\chi,1)\)  \(\approx\)  \(1.002698698 + 0.01203219186i\)
\(L(1,\chi)\)  \(\approx\)  \(1.002698698 + 0.01203219186i\)

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

−31.119345800830812067339303635126, −30.39883770389528582097887302939, −29.28245762763062913806365275976, −27.90980369953970360685359540634, −27.01245545162468757232465720899, −26.26816374965476859381224275339, −25.36088189018762385266954973220, −23.394458859052862334893841779892, −22.239561785113222393484353028037, −21.36254921243849200169608963666, −20.393196028298884408641031088529, −19.15341074706601018623725108930, −18.674096590188287876530922457398, −17.03618584715542267861300445081, −15.57894174254666558758283981415, −14.31949230308043821784798757193, −13.42732391816309275199519676114, −11.67919097708322044763377292873, −10.70887303491955109306917120725, −9.60618238131905646650268917638, −8.50199632430993164355383741449, −7.12405617327406251662328325077, −4.602635637096723602231467780023, −3.39294880185787643734756846901, −2.282248912904032642846602661801, 1.29819979742838519096564895388, 3.758570455065874744064063042460, 5.38909528559575485125783434717, 6.88583560842524454803895712761, 8.25659010312455200290410986619, 8.68645455875435743786075968443, 10.23452340510790345656292173913, 12.506948868296037870440631061558, 13.258392005195958396966905611274, 14.625437373363294451467811832797, 15.53454283332391589832045452979, 16.8084491102901441329411572682, 17.88978956433101531855604445518, 19.1306803698067717449662233125, 19.956773815534323803331065623663, 21.25212008651949262988401973597, 23.1777963179850444887877354605, 23.83106560132621771141941927429, 25.0338030058481403668120072804, 25.4314766272151304582807867518, 26.83731241853718166219659007596, 27.73767545710150054447725860504, 28.92822377737561451560139256188, 30.42443348884388036841996546001, 31.538222400468231143810899208383

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