Properties

Degree 1
Conductor 53
Sign $-0.502 - 0.864i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.502 - 0.864i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.502 - 0.864i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(53\)
\( \varepsilon \)  =  $-0.502 - 0.864i$
motivic weight  =  \(0\)
character  :  $\chi_{53} (25, \cdot )$
Sato-Tate  :  $\mu(26)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 53,\ (0:\ ),\ -0.502 - 0.864i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4942964076 - 0.8585990914i$
$L(\frac12,\chi)$  $\approx$  $0.4942964076 - 0.8585990914i$
$L(\chi,1)$  $\approx$  0.8133416769 - 0.7752761395i
$L(1,\chi)$  $\approx$  0.8133416769 - 0.7752761395i

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

−33.5690466842908764528978597488, −32.72068679895308000751795386298, −32.09686477223292375670052628643, −30.46644680163664179410804505774, −29.28663442264638188395562666336, −27.6755439078865654590571460940, −26.68238894604723656687977019349, −25.75146383500437438682557192686, −24.843604028920779935072194984567, −23.01757576099303267425030499320, −22.48025069305149336592311702594, −21.30405084722324793578895080028, −20.1088044690912189099508509835, −17.87843481501607267158272590693, −17.06153279856835297012959869397, −16.09322380539395313065681109499, −14.61519535182024487728919763363, −13.9577258005833326767507671423, −12.311401609437040617332018215550, −10.1445740632411993613359814304, −9.5167282833699589851414816214, −7.54369421658015629235940997436, −6.06994924182853995673295391877, −4.82616551079987322049069178809, −3.364494201405403378877903230574, 1.59086792655426903007158726037, 2.90284628429741877190039083080, 5.34027812958640334224853391224, 6.32592816031808680715932573425, 8.653494217623241520566964026462, 9.76873689789944420597201463295, 11.588134891999928449498507833407, 12.44101513902664073837304219754, 13.59440310530017895526567896281, 14.49579403393510004593269455662, 16.80051032042339450821259458419, 18.12596653311923533034341657850, 18.96351215139006536375939132505, 20.04485293876706227893470354243, 21.69007142795959756327677567039, 22.16985585030938049855571972711, 23.875400835595583669746341273461, 24.6727320096329733134687688833, 25.97968191425698754028868301535, 27.83929664416747383175947146360, 28.87848810390536285686267240302, 29.45778471874089473606031779075, 30.51173379889316199027408536748, 31.712454315860223450847072097756, 32.56261184463008714032850764383

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