Properties

Degree 1
Conductor $ 7^{2} $
Sign $0.991 + 0.127i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.955 − 0.294i)2-s + (0.365 + 0.930i)3-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (0.623 + 0.781i)6-s + (0.623 − 0.781i)8-s + (−0.733 + 0.680i)9-s + (−0.988 + 0.149i)10-s + (−0.733 − 0.680i)11-s + (0.826 + 0.563i)12-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s,χ)  = 1  + (0.955 − 0.294i)2-s + (0.365 + 0.930i)3-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (0.623 + 0.781i)6-s + (0.623 − 0.781i)8-s + (−0.733 + 0.680i)9-s + (−0.988 + 0.149i)10-s + (−0.733 − 0.680i)11-s + (0.826 + 0.563i)12-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.991 + 0.127i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.991 + 0.127i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(49\)    =    \(7^{2}\)
\( \varepsilon \)  =  $0.991 + 0.127i$
motivic weight  =  \(0\)
character  :  $\chi_{49} (44, \cdot )$
Sato-Tate  :  $\mu(21)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 49,\ (0:\ ),\ 0.991 + 0.127i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.330637733 + 0.08542977694i$
$L(\frac12,\chi)$  $\approx$  $1.330637733 + 0.08542977694i$
$L(\chi,1)$  $\approx$  1.497376326 + 0.05725960253i
$L(1,\chi)$  $\approx$  1.497376326 + 0.05725960253i

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

−33.93499038174627916039788289045, −32.30204312631567978726805372452, −31.40827476801357759521553955370, −30.58527771086907849952575903195, −29.74375058022985343423799112878, −28.297584695005528973526270020285, −26.48356823001286240713795612857, −25.43298148456378234145277402903, −24.29733773382151664454051526336, −23.37208599888442852978424003689, −22.59185625325205472918133634408, −20.720027424964980001405825136866, −19.854610007863292780674076329135, −18.52220622113332790883986972558, −16.95572976650942287513041571164, −15.3108302277856568318467021245, −14.61942841624935926472337733664, −12.90441522736308064912510543037, −12.38243410543024528483521183331, −10.83643171398057255273269501579, −8.12880084662476751108962297903, −7.439738180433075531259372471918, −5.90283386994892023595712208666, −3.998925599507717937789965092641, −2.47447128123156755365308935188, 2.850066632833490143980450551563, 4.12766929000533809832009000494, 5.2691573648571203008934620832, 7.37089748447892913034812335055, 9.126378929569324488787221386787, 10.831310679545591426563683962569, 11.672136512620887195363444365045, 13.360778851645792299025170619082, 14.59928258722468310821515956400, 15.72087356483794533645171396174, 16.43110541367687626713917002758, 19.01832151244170497590172326435, 19.97545657285596008758833549878, 21.07309241848162123565790824117, 21.9992314781284342786919297998, 23.28289606707130084029019993951, 24.20962632930678948764400427522, 25.7888013412687004353388984857, 27.03605971720192340642774216215, 28.13310395481958255798422796515, 29.330513938849831961941023226205, 30.940388947339481295276986115702, 31.58921922952435982852188521085, 32.34501179660431471401620203777, 33.70574318892804685853862949219

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