Properties

Degree 1
Conductor 41
Sign $-0.843 + 0.537i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.587 − 0.809i)2-s + (0.707 − 0.707i)3-s + (−0.309 + 0.951i)4-s + (−0.951 − 0.309i)5-s + (−0.987 − 0.156i)6-s + (−0.987 + 0.156i)7-s + (0.951 − 0.309i)8-s i·9-s + (0.309 + 0.951i)10-s + (−0.891 − 0.453i)11-s + (0.453 + 0.891i)12-s + (−0.156 + 0.987i)13-s + (0.707 + 0.707i)14-s + (−0.891 + 0.453i)15-s + (−0.809 − 0.587i)16-s + (−0.453 + 0.891i)17-s + ⋯
L(s,χ)  = 1  + (−0.587 − 0.809i)2-s + (0.707 − 0.707i)3-s + (−0.309 + 0.951i)4-s + (−0.951 − 0.309i)5-s + (−0.987 − 0.156i)6-s + (−0.987 + 0.156i)7-s + (0.951 − 0.309i)8-s i·9-s + (0.309 + 0.951i)10-s + (−0.891 − 0.453i)11-s + (0.453 + 0.891i)12-s + (−0.156 + 0.987i)13-s + (0.707 + 0.707i)14-s + (−0.891 + 0.453i)15-s + (−0.809 − 0.587i)16-s + (−0.453 + 0.891i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.843 + 0.537i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.843 + 0.537i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $-0.843 + 0.537i$
motivic weight  =  \(0\)
character  :  $\chi_{41} (35, \cdot )$
Sato-Tate  :  $\mu(40)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 41,\ (1:\ ),\ -0.843 + 0.537i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1219312920 - 0.4178552160i$
$L(\frac12,\chi)$  $\approx$  $-0.1219312920 - 0.4178552160i$
$L(\chi,1)$  $\approx$  0.4436780800 - 0.4009362910i
$L(1,\chi)$  $\approx$  0.4436780800 - 0.4009362910i

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

−35.406464564821670214858798431543, −34.1515344814477609716112608959, −33.061297550159361463909242657345, −31.91659049226436395498116774110, −31.1940386277135987729830534233, −29.17725840781471748257536724029, −27.67962623124966842394376939521, −26.934807592616775482069421960459, −25.92539815870658932707496321247, −25.029816001442385631528986902337, −23.293384657765226339493267257476, −22.484523219304329762053687016496, −20.3644300843454339832548743317, −19.50504117845041949781200446218, −18.32963977090478960401120195788, −16.42038983523539110685971545404, −15.64179510853793667768285502484, −14.754153053438757591589535073252, −13.12243901834824771312728433075, −10.7180341930161342724277016665, −9.69294072519639419169968340872, −8.21322079413197936365929003854, −7.13382404571849735863605510848, −5.01594336548300169329515616979, −3.19950903700724516860825017497, 0.293532809634490773928706235576, 2.50989426397282004739210628317, 3.92349027333154157023698506619, 6.97283483782186381511856246750, 8.30713561567245405467197362873, 9.31823639179916689025054978198, 11.191817851997362377627940504492, 12.5692596413842919198890408649, 13.338412325874074498012823896, 15.42483345634959340782516027988, 16.83508781948868701368519359784, 18.641609962778824395701553121694, 19.24682565041557751171234651163, 20.1605235055494734008123874095, 21.5047886013485078255870753193, 23.17129473631257978585631953432, 24.42062730162095619439752861984, 26.11349627732762239487832314497, 26.536062580776336151174579627766, 28.33422156252420842532061743533, 29.108712065001454156805144345129, 30.47368025628943931443543000512, 31.38330886740571678451778139691, 32.151916305722126442806008156511, 34.6547379321028542429700857192

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