Properties

Degree 1
Conductor $ 3^{2} \cdot 7 $
Sign $-0.805 + 0.592i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s − 5-s − 8-s + (−0.5 + 0.866i)10-s − 11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s − 23-s + 25-s + (0.5 + 0.866i)26-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s − 5-s − 8-s + (−0.5 + 0.866i)10-s − 11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s − 23-s + 25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.805 + 0.592i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.805 + 0.592i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(63\)    =    \(3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.805 + 0.592i$
motivic weight  =  \(0\)
character  :  $\chi_{63} (2, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 63,\ (1:\ ),\ -0.805 + 0.592i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.1523427185 - 0.4637776184i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.1523427185 - 0.4637776184i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6374146630 - 0.4694082409i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6374146630 - 0.4694082409i\)

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

−32.5609969293651311198500232679, −31.77382675878824807691733348230, −30.86482576814273861475276657861, −29.78537853009936930716878815426, −28.032459431641616801888324061593, −27.01013465197015233988404774550, −26.04506879730947531674930544654, −24.8330744310206676912662106752, −23.66250953355354062153317323445, −23.05074015715147677869480646583, −21.76339284304054345972240506932, −20.48038571443826156997941289305, −19.01584233836184106453203757147, −17.73561260779734798290115972659, −16.40609984630526446451048265662, −15.45918312962319185372878980082, −14.53234408657870888604843767065, −12.94746660762414333181602887927, −12.072478763352504260770963856581, −10.31351580554874013154419967473, −8.26901164242919121145894060990, −7.65923530927053757782151187933, −5.97673082023526117967730544775, −4.56365513714701777682869607341, −3.17955636089342400003914033975, 0.21101117689275705024907535069, 2.47227280735772256264480903533, 3.997194976488549994666400337440, 5.22726535841865591825911967903, 7.207873796731088656840733616976, 8.86147525634911443535607368494, 10.36209178411306790975518471829, 11.543492680274314995541217001905, 12.464404210585805112806457952090, 13.821245985967779568047736776893, 15.0667842415723153246067225999, 16.22068045351388620290112126219, 18.14794544565383608713697534769, 19.144332429991054335629986971517, 20.11116834892003220421090614301, 21.221727188370206708627332744028, 22.39247429235318845127978095058, 23.550537789947449390756680071386, 24.15262341526251445917825537758, 26.13825319004199676647898111630, 27.273036213713512915670246896984, 28.22359352980773594430243947320, 29.28868487745808144757263928577, 30.42010497109819597981691435296, 31.50430152621943434703189349017

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