Properties

Degree 1
Conductor 139
Sign $0.817 + 0.576i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.648 + 0.761i)2-s + (0.613 − 0.789i)3-s + (−0.158 − 0.987i)4-s + (0.538 + 0.842i)5-s + (0.203 + 0.979i)6-s + (0.746 + 0.665i)7-s + (0.854 + 0.519i)8-s + (−0.247 − 0.968i)9-s + (−0.990 − 0.136i)10-s + (−0.974 − 0.225i)11-s + (−0.877 − 0.480i)12-s + (0.377 + 0.926i)13-s + (−0.990 + 0.136i)14-s + (0.995 + 0.0909i)15-s + (−0.949 + 0.313i)16-s + (0.113 − 0.993i)17-s + ⋯
L(s,χ)  = 1  + (−0.648 + 0.761i)2-s + (0.613 − 0.789i)3-s + (−0.158 − 0.987i)4-s + (0.538 + 0.842i)5-s + (0.203 + 0.979i)6-s + (0.746 + 0.665i)7-s + (0.854 + 0.519i)8-s + (−0.247 − 0.968i)9-s + (−0.990 − 0.136i)10-s + (−0.974 − 0.225i)11-s + (−0.877 − 0.480i)12-s + (0.377 + 0.926i)13-s + (−0.990 + 0.136i)14-s + (0.995 + 0.0909i)15-s + (−0.949 + 0.313i)16-s + (0.113 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.817 + 0.576i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.817 + 0.576i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(139\)
\( \varepsilon \)  =  $0.817 + 0.576i$
motivic weight  =  \(0\)
character  :  $\chi_{139} (7, \cdot )$
Sato-Tate  :  $\mu(69)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 139,\ (0:\ ),\ 0.817 + 0.576i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.019869602 + 0.3236023975i$
$L(\frac12,\chi)$  $\approx$  $1.019869602 + 0.3236023975i$
$L(\chi,1)$  $\approx$  1.001658906 + 0.2257161215i
$L(1,\chi)$  $\approx$  1.001658906 + 0.2257161215i

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

−28.126150864524528786232960206311, −27.52204577149853516880320372564, −26.43765144494909353163859912290, −25.72536183837466846695910847927, −24.6930965149160201161462047753, −23.267386996102670497920917268652, −21.72453650045880590196860109644, −21.13599215748935617913205939567, −20.30564358831127812792005196915, −19.77692456514969001106529132795, −18.12865862493154097369854977115, −17.31673312189779598192642980112, −16.33609120667692777980108530789, −15.20365659951080279025787127061, −13.575058085151831504339540551219, −13.06676110450190398719369856800, −11.385467529705183591696320677271, −10.34078836642340838472321304667, −9.65122406290527003705880791504, −8.33474981087587966451204472361, −7.80951482286307131768031022974, −5.27473848789193854228361109492, −4.23646486539278225330008757690, −2.84826216081643360075193154551, −1.4022832704378011803198696479, 1.63578336842254283067396660257, 2.78970241800130730123324400297, 5.17955676299367176246874895155, 6.365175922532507579939037802, 7.32601216777571603082933605472, 8.35208701978195629393107797102, 9.314713714462719021560705670521, 10.609878444005561898777757265288, 11.877563788551183368680357086155, 13.64796599925520770862110042746, 14.21803990972927001837974613488, 15.125770350066310073947484629042, 16.33186083274742893223579758385, 17.902385268936554323716216461860, 18.36471628967476117715009863001, 18.8838154715277671913194587015, 20.36503717865789547860549822538, 21.42192499266609738058136162717, 22.95011723941905042457595672555, 23.864667355984321926577157456, 24.843440819042415919790073786786, 25.450604762059465885721955211584, 26.45673237875827083801670932237, 27.07785023493026008307503023959, 28.77120437754632197270261647683

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