Properties

Degree 1
Conductor 67
Sign $0.884 - 0.466i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.327 + 0.945i)2-s + (0.415 − 0.909i)3-s + (−0.786 − 0.618i)4-s + (−0.959 − 0.281i)5-s + (0.723 + 0.690i)6-s + (0.981 − 0.189i)7-s + (0.841 − 0.540i)8-s + (−0.654 − 0.755i)9-s + (0.580 − 0.814i)10-s + (0.723 − 0.690i)11-s + (−0.888 + 0.458i)12-s + (0.0475 − 0.998i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.235 + 0.971i)16-s + (−0.786 + 0.618i)17-s + ⋯
L(s,χ)  = 1  + (−0.327 + 0.945i)2-s + (0.415 − 0.909i)3-s + (−0.786 − 0.618i)4-s + (−0.959 − 0.281i)5-s + (0.723 + 0.690i)6-s + (0.981 − 0.189i)7-s + (0.841 − 0.540i)8-s + (−0.654 − 0.755i)9-s + (0.580 − 0.814i)10-s + (0.723 − 0.690i)11-s + (−0.888 + 0.458i)12-s + (0.0475 − 0.998i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.235 + 0.971i)16-s + (−0.786 + 0.618i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.884 - 0.466i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.884 - 0.466i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(67\)
\( \varepsilon \)  =  $0.884 - 0.466i$
motivic weight  =  \(0\)
character  :  $\chi_{67} (26, \cdot )$
Sato-Tate  :  $\mu(33)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 67,\ (0:\ ),\ 0.884 - 0.466i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7492964729 - 0.1855687512i$
$L(\frac12,\chi)$  $\approx$  $0.7492964729 - 0.1855687512i$
$L(\chi,1)$  $\approx$  0.8679368596 - 0.04745357172i
$L(1,\chi)$  $\approx$  0.8679368596 - 0.04745357172i

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

−31.553926166091846326295322124004, −31.08992670347259290615659363115, −30.19978082461648686850430451590, −28.41416688227316714758633361483, −27.739587509479179799422993459317, −26.851165978601509017305126807920, −26.1147300693187139120954056606, −24.373712540302355235373596821761, −22.73025261994922652641989991148, −21.971999294270903012792891492328, −20.67438670421583788399339048724, −20.03335999326080824786445752801, −18.876009220735533232926290405563, −17.57746501570574829763625332973, −16.18939253699611073369642725726, −14.85593208080427502794458010296, −13.84752050474167424910229471070, −11.7221656927878087776155340966, −11.37859362603248039669127172114, −9.81583390770055089072448434894, −8.74382958658421988139989894188, −7.56304947628579913487622443699, −4.674471022641845054633186923835, −3.91391010053088313586512603635, −2.22679579646755516324469845772, 1.16590467472096318942326007714, 3.841669938420712557298148518155, 5.59053572189227000832371528448, 7.162561215364008887270241104565, 8.08532610699734307881781029376, 8.850189117287163057577608172639, 10.99518796060256176299056814761, 12.42098114260366220419814346816, 13.85162479774743859882121880286, 14.74762793533473119096764898239, 15.95114720897401618100692643987, 17.37926303479539347700597782434, 18.22289108831875520451155591975, 19.48575700057573767401990726051, 20.24210935441086887925905114445, 22.35666136593634562160753361710, 23.64103166219974861653850395100, 24.28132924695124794978654434831, 25.00771348670756018758762659672, 26.48847998507363446992575999382, 27.288054611172430946131287434062, 28.35715728845352307740099403615, 30.03435867599763825455200319921, 30.98404049496636086906507140131, 31.91180145287161943687017149310

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