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  :  $\chi_{71} (64, \cdot )$
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

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

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