Properties

Degree 1
Conductor 71
Sign $-0.563 - 0.826i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.858 − 0.512i)2-s + (−0.691 − 0.722i)3-s + (0.473 − 0.880i)4-s + (−0.809 − 0.587i)5-s + (−0.963 − 0.266i)6-s + (−0.995 + 0.0896i)7-s + (−0.0448 − 0.998i)8-s + (−0.0448 + 0.998i)9-s + (−0.995 − 0.0896i)10-s + (0.936 + 0.351i)11-s + (−0.963 + 0.266i)12-s + (0.936 − 0.351i)13-s + (−0.809 + 0.587i)14-s + (0.134 + 0.990i)15-s + (−0.550 − 0.834i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s,χ)  = 1  + (0.858 − 0.512i)2-s + (−0.691 − 0.722i)3-s + (0.473 − 0.880i)4-s + (−0.809 − 0.587i)5-s + (−0.963 − 0.266i)6-s + (−0.995 + 0.0896i)7-s + (−0.0448 − 0.998i)8-s + (−0.0448 + 0.998i)9-s + (−0.995 − 0.0896i)10-s + (0.936 + 0.351i)11-s + (−0.963 + 0.266i)12-s + (0.936 − 0.351i)13-s + (−0.809 + 0.587i)14-s + (0.134 + 0.990i)15-s + (−0.550 − 0.834i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.563 - 0.826i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.563 - 0.826i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $-0.563 - 0.826i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (10, \cdot )$
Sato-Tate  :  $\mu(35)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (0:\ ),\ -0.563 - 0.826i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4732078119 - 0.8950102589i$
$L(\frac12,\chi)$  $\approx$  $0.4732078119 - 0.8950102589i$
$L(\chi,1)$  $\approx$  0.8577348802 - 0.7205995562i
$L(1,\chi)$  $\approx$  0.8577348802 - 0.7205995562i

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

−32.39828485040108221611557137433, −31.15196279100106452233606213171, −30.06626314811157784422628535629, −28.975186830181670646158736788918, −27.59332852929106784938673218091, −26.491434358693756607632256673925, −25.68174351676052905849210894738, −24.0408666563966570425680970225, −23.00251469888533277112526010703, −22.504267497528462547354398511006, −21.4782286909512139676369317095, −20.12913786719402155772185090499, −18.68008033691442576691631263708, −16.90289173744616082955921770640, −16.21577340015281566659971038336, −15.25010360536927764937588813809, −14.159664481822248844405997729734, −12.4809886017335507961152741556, −11.56964100476287674895262994976, −10.36109399982570684507443036130, −8.5138309412885460377494696581, −6.68911052415497478836036524761, −6.01775519316817291415837650346, −4.06579946118368735705459201803, −3.485356020490125895473834653133, 1.11391127930369398696664897707, 3.24974655157507772381136511187, 4.76823621762235705251646180147, 6.14614222746211719757850112390, 7.27370531334757447162049620228, 9.30402566431042365181749123521, 11.03238994073998102683527729593, 11.98818084750195147579091271617, 12.78789548071733781029131528327, 13.81896843702951481942506119549, 15.634805321421686655063365016483, 16.39247767224229743619463923465, 18.11583538627741893841930874565, 19.51564474088557698558017762567, 19.91517296082412635926050543967, 21.62914271381529403184153099933, 22.9454892412738457443376113571, 23.18550963814036982529096614348, 24.51298411398796231053162846998, 25.37519231455654217604963836775, 27.61679402363694449772124870485, 28.276230379843738043666286152711, 29.2892456426333973147924525970, 30.26572411357450062331934712308, 31.17325489188524545912344591687

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