Properties

Degree 1
Conductor 53
Sign $0.974 + 0.223i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.464 + 0.885i)2-s + (−0.935 − 0.354i)3-s + (−0.568 − 0.822i)4-s + (−0.663 + 0.748i)5-s + (0.748 − 0.663i)6-s + (−0.885 − 0.464i)7-s + (0.992 − 0.120i)8-s + (0.748 + 0.663i)9-s + (−0.354 − 0.935i)10-s + (0.970 − 0.239i)11-s + (0.239 + 0.970i)12-s + (0.568 − 0.822i)13-s + (0.822 − 0.568i)14-s + (0.885 − 0.464i)15-s + (−0.354 + 0.935i)16-s + (−0.120 + 0.992i)17-s + ⋯
L(s,χ)  = 1  + (−0.464 + 0.885i)2-s + (−0.935 − 0.354i)3-s + (−0.568 − 0.822i)4-s + (−0.663 + 0.748i)5-s + (0.748 − 0.663i)6-s + (−0.885 − 0.464i)7-s + (0.992 − 0.120i)8-s + (0.748 + 0.663i)9-s + (−0.354 − 0.935i)10-s + (0.970 − 0.239i)11-s + (0.239 + 0.970i)12-s + (0.568 − 0.822i)13-s + (0.822 − 0.568i)14-s + (0.885 − 0.464i)15-s + (−0.354 + 0.935i)16-s + (−0.120 + 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.974 + 0.223i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.974 + 0.223i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(53\)
\( \varepsilon \)  =  $0.974 + 0.223i$
motivic weight  =  \(0\)
character  :  $\chi_{53} (3, \cdot )$
Sato-Tate  :  $\mu(52)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 53,\ (1:\ ),\ 0.974 + 0.223i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6520215181 + 0.07368321742i$
$L(\frac12,\chi)$  $\approx$  $0.6520215181 + 0.07368321742i$
$L(\chi,1)$  $\approx$  0.5616792506 + 0.1337864386i
$L(1,\chi)$  $\approx$  0.5616792506 + 0.1337864386i

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

−32.86758732305956094774794649439, −31.82914700262738242036535887014, −30.64848305321330307529042223381, −29.16245865348261517181041213228, −28.53795497417586848549755560256, −27.64408509698383129879425741715, −26.71857324314472984500184666599, −25.13809874315550571974268085954, −23.45434361877314586081042800659, −22.466112294975764040193574027125, −21.47326440306048724280080607217, −20.15812724219843410397963382648, −19.11377607204480975720562022097, −17.81121938741046804410668036062, −16.54477872593990623225208791310, −15.81576096753828542963953671610, −13.37223934259121379014288757267, −11.93217260050464151694703400533, −11.61773906104938398744835432012, −9.75565855673807396025602286463, −8.93322735665775311777423070736, −6.8863799965698421056089660932, −4.87665022653219787225984091157, −3.57310421837356414602630288779, −0.99556171507493885097412665946, 0.70348090182065996456245498293, 3.97104035173723032165264670571, 6.01472429153184783679318541306, 6.781804532812151577860782509608, 8.05985051822510659142886274130, 10.029648352030280323598427784092, 11.04139414699101542300972152580, 12.71360161385347455822517850905, 14.21988611405289132978131104985, 15.68716350996335423344670479012, 16.56759759432066821041029934414, 17.74189024223082961955061900653, 18.85609878790610369991124434629, 19.719799261559173669966714358536, 22.39247553119063226257846503581, 22.773662708426880937653882247186, 23.86471599900087238612202211961, 25.068562670967316639857521143114, 26.366577831452915428852861257521, 27.3346466799053395884772586749, 28.34414456155551977549656150713, 29.61899536161661579903965319127, 30.732690730442239783999524942028, 32.51733017242163942417733347521, 33.21033919235834912210207186122

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