Properties

Degree 1
Conductor 67
Sign $-0.0427 + 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.235 − 0.971i)2-s + (0.142 − 0.989i)3-s + (−0.888 + 0.458i)4-s + (−0.415 − 0.909i)5-s + (−0.995 + 0.0950i)6-s + (−0.723 + 0.690i)7-s + (0.654 + 0.755i)8-s + (−0.959 − 0.281i)9-s + (−0.786 + 0.618i)10-s + (0.995 + 0.0950i)11-s + (0.327 + 0.945i)12-s + (−0.981 − 0.189i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (0.580 − 0.814i)16-s + (−0.888 − 0.458i)17-s + ⋯
L(s,χ)  = 1  + (−0.235 − 0.971i)2-s + (0.142 − 0.989i)3-s + (−0.888 + 0.458i)4-s + (−0.415 − 0.909i)5-s + (−0.995 + 0.0950i)6-s + (−0.723 + 0.690i)7-s + (0.654 + 0.755i)8-s + (−0.959 − 0.281i)9-s + (−0.786 + 0.618i)10-s + (0.995 + 0.0950i)11-s + (0.327 + 0.945i)12-s + (−0.981 − 0.189i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (0.580 − 0.814i)16-s + (−0.888 − 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.0427 + 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.0427 + 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(67\)
\( \varepsilon \)  =  $-0.0427 + 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{67} (31, \cdot )$
Sato-Tate  :  $\mu(66)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 67,\ (1:\ ),\ -0.0427 + 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.2311552968 - 0.2412487050i$
$L(\frac12,\chi)$  $\approx$  $-0.2311552968 - 0.2412487050i$
$L(\chi,1)$  $\approx$  0.3840667339 - 0.4842414348i
$L(1,\chi)$  $\approx$  0.3840667339 - 0.4842414348i

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

−32.843049895361991082389289249924, −31.807333348019920142889475359680, −30.69396082828562750692487569978, −29.08402570202743110676650927914, −27.67930396567657459577220535877, −26.68425359846695469343848675031, −26.36392822302508157171971880355, −25.10763855079034886061246268163, −23.65367432177171323038390276894, −22.376590023635770906615249321751, −22.119714462490801095149372034994, −19.894765920910170106989730107180, −19.21877079195790663560318039087, −17.46325254550229095499590335717, −16.60982246934517831021452366821, −15.408451816307563738155821101296, −14.66200430994160360107630338381, −13.52674880561739693932485684495, −11.3020863155409392144627707716, −10.02463076962541092955836134416, −9.123956886897535805268150722, −7.42539972206907012366073653375, −6.38174774208723366470157603529, −4.58125460744182501495840348828, −3.40413140127530842370900574193, 0.182067099169218770077179162878, 1.793396757400814258191200927382, 3.3672689823904322947849885081, 5.21609314381678136842506572581, 7.174083105132360629043470285563, 8.69126521867627102298773595704, 9.4122936263081112880217685982, 11.510370882560545135976879138502, 12.37539977735821755507827182933, 13.025015845638366479417315152689, 14.51934978706317377939586335051, 16.47875578980638668847243351777, 17.596790537324642108645053965938, 18.83767058968774737566047991601, 19.70616667299595254338256379778, 20.375311103543809200757359879127, 22.074455703962986883259130326612, 22.94711788262127004554634182815, 24.46000186949901481000853823899, 25.22720196263293114390647521192, 26.80183719069829829439603511857, 27.94871634095372866626158018997, 28.98106917782230980248264300992, 29.55208264343025137197584251854, 31.0841039589603772478608382315

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