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Properties

Label 1-1792-1792.723-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.809 + 0.587i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (−0.528 − 0.849i)3-s + (0.162 − 0.986i)5-s + (−0.442 + 0.896i)9-s + (0.683 + 0.729i)11-s + (0.634 + 0.773i)13-s + (−0.923 + 0.382i)15-s + (−0.793 + 0.608i)17-s + (−0.412 − 0.910i)19-s + (−0.321 − 0.946i)23-s + (−0.946 − 0.321i)25-s + (0.995 − 0.0980i)27-s + (−0.956 + 0.290i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (−0.812 + 0.582i)37-s + ⋯
L(s)  = 1  + (−0.528 − 0.849i)3-s + (0.162 − 0.986i)5-s + (−0.442 + 0.896i)9-s + (0.683 + 0.729i)11-s + (0.634 + 0.773i)13-s + (−0.923 + 0.382i)15-s + (−0.793 + 0.608i)17-s + (−0.412 − 0.910i)19-s + (−0.321 − 0.946i)23-s + (−0.946 − 0.321i)25-s + (0.995 − 0.0980i)27-s + (−0.956 + 0.290i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (−0.812 + 0.582i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.809 + 0.587i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.809 + 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8286652064 + 0.2690488415i\)
\(L(\frac12)\) \(\approx\) \(0.8286652064 + 0.2690488415i\)
\(L(1)\) \(\approx\) \(0.7918860972 - 0.2667182823i\)
\(L(1)\) \(\approx\) \(0.7918860972 - 0.2667182823i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.528 - 0.849i)T \)
5 \( 1 + (0.162 - 0.986i)T \)
11 \( 1 + (0.683 + 0.729i)T \)
13 \( 1 + (0.634 + 0.773i)T \)
17 \( 1 + (-0.793 + 0.608i)T \)
19 \( 1 + (-0.412 - 0.910i)T \)
23 \( 1 + (-0.321 - 0.946i)T \)
29 \( 1 + (-0.956 + 0.290i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (-0.812 + 0.582i)T \)
41 \( 1 + (0.195 - 0.980i)T \)
43 \( 1 + (0.471 + 0.881i)T \)
47 \( 1 + (0.991 - 0.130i)T \)
53 \( 1 + (0.729 - 0.683i)T \)
59 \( 1 + (-0.352 - 0.935i)T \)
61 \( 1 + (-0.0327 - 0.999i)T \)
67 \( 1 + (-0.849 + 0.528i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (0.0654 + 0.997i)T \)
79 \( 1 + (0.608 - 0.793i)T \)
83 \( 1 + (0.0980 - 0.995i)T \)
89 \( 1 + (0.659 + 0.751i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−19.967710735328603181427263511745, −19.229285520676713613961866617242, −18.233922655807421876316789062231, −17.83721930660289806399236384059, −16.96428795610294983209549346775, −16.24668237808516091561811573076, −15.4585932195654185383695946701, −14.95599150161121141346670277332, −14.04853580128345641955246911455, −13.49534366591470452685046437708, −12.22143267621476015553010811598, −11.54976528984713318162745803832, −10.73125972388468922476553992789, −10.49018657066997587012607050909, −9.37960672324810618726238926409, −8.8243872069441425599388036780, −7.673497064108371201087814860841, −6.750915509870941159826202411902, −5.89411406853542072581800579116, −5.54610841420816758066092560146, −4.15067339109264243657328442695, −3.59740436328759096177868858779, −2.82045155923227875522092223957, −1.50411246169563061430744006131, −0.21247262160559125416036318491, 0.77217164812697321684751061292, 1.768686195405605854365927267230, 2.23907341394689933631420042821, 3.98213962838442856898892604259, 4.551768231833355254363726217912, 5.50391320698336300419992075532, 6.379651117175075039405683720523, 6.89526122197957007561564825331, 7.91592777279334085225899310952, 8.79933284874312822846898882694, 9.235999088088034824169850081297, 10.467524274398706887900215405704, 11.29262961404835948864056260397, 11.94931240378694091587042018819, 12.72063418188080282802158808060, 13.19460359205291363716532064239, 13.94756783811908273366548148909, 14.87588433645488663460812368808, 15.87105027599218888397389876435, 16.59441318947791857221259492910, 17.30504775190902600118612605108, 17.63479291474735322783434153340, 18.687479987970958653819839819878, 19.2686540097745619125940163726, 20.16287652895853111123841844324

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