Properties

Degree 1
Conductor $ 2^{3} \cdot 7 $
Sign $0.895 + 0.444i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s − 13-s + 15-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + ⋯
L(s,χ)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s − 13-s + 15-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.895 + 0.444i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.895 + 0.444i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(56\)    =    \(2^{3} \cdot 7\)
\( \varepsilon \)  =  $0.895 + 0.444i$
motivic weight  =  \(0\)
character  :  $\chi_{56} (53, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 56,\ (0:\ ),\ 0.895 + 0.444i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.015749404 + 0.2379507569i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.015749404 + 0.2379507569i\)
\(L(\chi,1)\)  \(\approx\)  \(1.144416812 + 0.1890773524i\)
\(L(1,\chi)\)  \(\approx\)  \(1.144416812 + 0.1890773524i\)

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

−32.878652151155462845550774415780, −31.6858212545645373072385833513, −30.54313326014748863362206707886, −29.711538087543605866881860291912, −28.88164856972576021512460322664, −26.96942614732156147411361702982, −26.14801703818197893491223395759, −24.93777312522420219237719763657, −24.119636756062502195205479758392, −22.626027887732602347617102642830, −21.59229649401529206561549190091, −20.019811095619857066711058797069, −18.99002657639154825336832760314, −18.067126830961526936755658746915, −16.83173382948516944677621157313, −14.82168300855820107816928743341, −14.13726659162291285573375640509, −12.86403045089641002680052356428, −11.47394543335907408927272641089, −9.9242602144380768221004404504, −8.421353673615397191135210682863, −7.02960834117298962590599781846, −5.93914354997256888939485106431, −3.42657634915339302822986887042, −1.979187922552712419455905670158, 2.286932854124725017211133938959, 4.275643771230588155053918649108, 5.34514023322726001166668039247, 7.5001379108129929193308950465, 9.22939246937515407603293390851, 9.697367100622115517238410527, 11.51874359844929761671569982875, 13.05736614935335163935275630588, 14.30072552858230477095100671407, 15.516200172712180204782466577034, 16.69457854557393395856180513184, 17.73802521766610363020169011157, 19.77054595273005960804550521510, 20.35104712810075107830630665127, 21.60931796040119349508270047159, 22.515172534202875348241627428914, 24.31433937438968621315748409359, 25.22428274137713926686550752869, 26.32609648903301844075867008414, 27.57475672962583227682316732842, 28.377901088678679940072714335614, 29.6928220809198263794187545187, 31.22745060950545181521565468473, 32.03544889421736177114128467462, 33.063039205310755923098035665845

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