Properties

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

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.222 − 0.974i)2-s + (−0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (−0.623 + 0.781i)6-s + (0.623 + 0.781i)8-s + (−0.222 + 0.974i)9-s + (−0.623 + 0.781i)10-s + (−0.222 − 0.974i)11-s + (0.900 + 0.433i)12-s + (0.222 + 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + (0.900 + 0.433i)17-s + 18-s − 19-s + ⋯
L(s,χ)  = 1  + (−0.222 − 0.974i)2-s + (−0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (−0.623 + 0.781i)6-s + (0.623 + 0.781i)8-s + (−0.222 + 0.974i)9-s + (−0.623 + 0.781i)10-s + (−0.222 − 0.974i)11-s + (0.900 + 0.433i)12-s + (0.222 + 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + (0.900 + 0.433i)17-s + 18-s − 19-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(49\)    =    \(7^{2}\)
\( \varepsilon \)  =  $0.127 + 0.991i$
motivic weight  =  \(0\)
character  :  $\chi_{49} (6, \cdot )$
Sato-Tate  :  $\mu(14)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 49,\ (1:\ ),\ 0.127 + 0.991i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1022807544 - 0.08993978631i$
$L(\frac12,\chi)$  $\approx$  $-0.1022807544 - 0.08993978631i$
$L(\chi,1)$  $\approx$  0.3465398303 - 0.3578308262i
$L(1,\chi)$  $\approx$  0.3465398303 - 0.3578308262i

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

−34.32132191791602344623166785396, −33.54442619454019371686546168076, −32.419662208635052530849446075536, −31.36048576416160121411052521903, −29.85405192250389225320542923475, −27.98815417174672432578005475061, −27.58172383647540370428405068348, −26.323572192553112713906426806820, −25.47682069789491921971331773244, −23.64666608843425817131494988756, −22.932980441360385057175933783970, −22.07045335029736531304572323858, −20.29903368798745191200525557741, −18.59648099722324866772376323050, −17.64854010516645228562248000575, −16.34951559465512605939900101800, −15.29601709252427400113956889433, −14.577539852530280407368602386910, −12.5062837505297389763323536249, −10.76048714437120514921038290753, −9.79019251964555466950469929460, −8.00884983912041497385627311933, −6.6511913232285671756233503405, −5.2235870049862019131840397596, −3.767150760966424750135916042970, 0.09468210403770850591718334042, 1.68366398596360998847434856293, 3.9255330430267476667620634680, 5.598758255829638354504644268362, 7.73790205128176591311583814908, 8.871511114139271950507747319070, 10.79358360510946526622781435530, 11.824299840207162009259677376349, 12.743774063369691765024512201331, 13.91567116801008849611805413120, 16.34025813678441105045203585633, 17.21969445983074282213740940504, 18.813445915931509027239848592783, 19.30796500154226098349333376649, 20.810956834655686726207779722600, 21.97489319626256060309273023807, 23.458746317477307567237136768192, 24.04831935121641523476395696938, 25.85121991079588928671589939455, 27.42159671426839651607334626352, 28.21032138635890293142640010271, 29.18972083112871472095794891490, 30.15188589711422934762287854353, 31.31821818457551084958004083870, 32.21869067791450474824643325273

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