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Properties

Label 1-840-840.797-r1-0-0
Degree $1$
Conductor $840$
Sign $-0.525 + 0.850i$
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  + 11-s − i·13-s + i·17-s − 19-s + i·23-s − 29-s − 31-s − i·37-s + 41-s + i·43-s + i·47-s − i·53-s + 59-s + 61-s − i·67-s + ⋯
L(s)  = 1  + 11-s − i·13-s + i·17-s − 19-s + i·23-s − 29-s − 31-s − i·37-s + 41-s + i·43-s + i·47-s − i·53-s + 59-s + 61-s − i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(90.2705\)
Root analytic conductor: \(90.2705\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 840,\ (1:\ ),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4849746343 + 0.8698526832i\)
\(L(\frac12)\) \(\approx\) \(0.4849746343 + 0.8698526832i\)
\(L(1)\) \(\approx\) \(0.9655990293 + 0.08706801016i\)
\(L(1)\) \(\approx\) \(0.9655990293 + 0.08706801016i\)

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 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 \)
23 \( 1 \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 + T \)
41 \( 1 \)
43 \( 1 - iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

−21.82017985491528137177564932831, −20.75736335992599597155537356745, −20.18668261496407599389286650390, −19.11317374018085585425491461107, −18.670036752027033257798035064586, −17.60819347564226494983050920937, −16.71368618167360776022304628970, −16.29879264338436298068996812078, −15.0371453568204275300642789148, −14.44824581269075586431076523305, −13.63930901379471422976419401537, −12.67251720905273047880767824947, −11.7910894367252571228735440224, −11.15860571548365585747839923930, −10.09532634571323333192392249994, −9.13664035048234410111287002487, −8.62136682918820698234618105787, −7.262213845427009726867335887944, −6.68291803626548647774343997434, −5.67887885674895382573717692703, −4.49834076547247798853338570587, −3.84140688398401283387171158195, −2.525594368434424959941052820659, −1.55908758636164462557025098763, −0.22325562861659992620306028288, 1.1618006269651495216150432531, 2.18178822279117112102783202312, 3.51235446921492781054326515033, 4.152438817382792775792578058384, 5.48081470334559414095938172402, 6.152411173668517899867139717574, 7.21780536953781146891244092686, 8.09476024340309120729844331584, 8.99969408558041406549974720482, 9.8175484928910116931256503535, 10.8423484625382623867343237382, 11.4487302287844157266661886824, 12.74332071834466655570172234650, 12.960409353700813089807506390501, 14.36576287419929062426802349811, 14.82163537640811535430833883220, 15.7288637423536897970794775925, 16.70777365571155420573495698361, 17.41909534300822004231907278030, 18.05130886819140091826211163345, 19.35524326399414988420301376871, 19.56420597534955474524971243253, 20.62599144931173212722726866707, 21.4575725271435527812371808167, 22.204530612944109580410528521711

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