Properties

Label 1-840-840.677-r1-0-0
Degree $1$
Conductor $840$
Sign $0.581 + 0.813i$
Analytic cond. $90.2705$
Root an. cond. $90.2705$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)11-s i·13-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 − 0.5i)23-s − 29-s + (0.5 − 0.866i)31-s + (−0.866 + 0.5i)37-s + 41-s i·43-s + (0.866 − 0.5i)47-s + (0.866 + 0.5i)53-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)61-s + (0.866 + 0.5i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)11-s i·13-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 − 0.5i)23-s − 29-s + (0.5 − 0.866i)31-s + (−0.866 + 0.5i)37-s + 41-s i·43-s + (0.866 − 0.5i)47-s + (0.866 + 0.5i)53-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)61-s + (0.866 + 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(90.2705\)
Root analytic conductor: \(90.2705\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 840,\ (1:\ ),\ 0.581 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.419846908 + 0.7305671897i\)
\(L(\frac12)\) \(\approx\) \(1.419846908 + 0.7305671897i\)
\(L(1)\) \(\approx\) \(1.008241375 + 0.06938860474i\)
\(L(1)\) \(\approx\) \(1.008241375 + 0.06938860474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.676714630873111087229067193767, −21.201824146346498408494891655877, −20.20146052348008169346929200892, −19.28895878862874038463123216322, −18.79965913769418573554721665354, −17.73845767655963701703738606366, −17.077160166417015797947800315480, −16.07073539879259296455627597326, −15.53961454584672704678514377104, −14.48130911156022707050612841761, −13.625813834973149946741041285359, −13.06008290319100893382207912298, −11.91447371810810779406202837928, −11.13123421371425411249282466677, −10.49625350780570463218791416044, −9.13265990010577123089590769142, −8.82053853333936563915868914781, −7.54362823133393184554470289683, −6.79147753125764199467731983019, −5.77560030713564689251259545479, −4.87269655302776978355124240617, −3.84034674922606737265136423817, −2.82956882666799970971341922314, −1.744598842914779703693959745755, −0.446508001518676856018559203956, 0.8282237958255442250184918830, 2.155801764800821458324117342133, 3.04354433989887468491379419463, 4.23919368139397678041332573886, 5.13058390610598137304188425474, 6.006298471349133659361774940, 7.17408000309204316141571068917, 7.79100939316839587226876741098, 8.83991336743908020148738500401, 9.77150022067394092030166946551, 10.51719944967586521138770952815, 11.395241309161247912693448544664, 12.42101762055983111798974212829, 13.043879444537855062314886242, 13.90508705464900918845182278942, 15.036090129464693658087344993, 15.42260319944684787228149334645, 16.45964245795700618683114968809, 17.32216272309894643213170357974, 18.100184083181217522750778452288, 18.71741519401011045184665198910, 19.81833134945952778468114744806, 20.54315679283748287034894936132, 20.99927254360297143423649827621, 22.37032742356959338113096875667

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