Properties

Label 1-1792-1792.1301-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.0245 - 0.999i$
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  + (0.773 + 0.634i)3-s + (0.290 + 0.956i)5-s + (0.195 + 0.980i)9-s + (0.0980 − 0.995i)11-s + (−0.956 − 0.290i)13-s + (−0.382 + 0.923i)15-s + (0.382 + 0.923i)17-s + (−0.471 − 0.881i)19-s + (0.555 − 0.831i)23-s + (−0.831 + 0.555i)25-s + (−0.471 + 0.881i)27-s + (−0.995 + 0.0980i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (−0.881 − 0.471i)37-s + ⋯
L(s)  = 1  + (0.773 + 0.634i)3-s + (0.290 + 0.956i)5-s + (0.195 + 0.980i)9-s + (0.0980 − 0.995i)11-s + (−0.956 − 0.290i)13-s + (−0.382 + 0.923i)15-s + (0.382 + 0.923i)17-s + (−0.471 − 0.881i)19-s + (0.555 − 0.831i)23-s + (−0.831 + 0.555i)25-s + (−0.471 + 0.881i)27-s + (−0.995 + 0.0980i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (−0.881 − 0.471i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0245 - 0.999i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.0245 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4989544703 - 0.4868561465i\)
\(L(\frac12)\) \(\approx\) \(0.4989544703 - 0.4868561465i\)
\(L(1)\) \(\approx\) \(1.132252991 + 0.3457636619i\)
\(L(1)\) \(\approx\) \(1.132252991 + 0.3457636619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.773 + 0.634i)T \)
5 \( 1 + (0.290 + 0.956i)T \)
11 \( 1 + (0.0980 - 0.995i)T \)
13 \( 1 + (-0.956 - 0.290i)T \)
17 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (-0.471 - 0.881i)T \)
23 \( 1 + (0.555 - 0.831i)T \)
29 \( 1 + (-0.995 + 0.0980i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.881 - 0.471i)T \)
41 \( 1 + (0.831 + 0.555i)T \)
43 \( 1 + (0.773 - 0.634i)T \)
47 \( 1 + (-0.923 + 0.382i)T \)
53 \( 1 + (-0.995 - 0.0980i)T \)
59 \( 1 + (-0.956 + 0.290i)T \)
61 \( 1 + (0.634 - 0.773i)T \)
67 \( 1 + (0.634 - 0.773i)T \)
71 \( 1 + (-0.195 + 0.980i)T \)
73 \( 1 + (-0.980 + 0.195i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (0.881 - 0.471i)T \)
89 \( 1 + (-0.555 - 0.831i)T \)
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

−20.36689240194428084404778604486, −19.36448380501429032949164571082, −19.00096950791100896330694344353, −17.90326461474029571135064003827, −17.35351476849468639507244852228, −16.66306678723180416169317056120, −15.679526634570265028596364759680, −14.88989546828569812732642369983, −14.24306187061048370827890749227, −13.47059999749505467969963411496, −12.724717101614457120653641726668, −12.23488527975346668658551769993, −11.53071574529799053280694663243, −10.00493330767913552928939814804, −9.53518690241294000702847151739, −8.95396859303001060013075209915, −7.86261337559647842081300314356, −7.465939808714403501565825446677, −6.52019074196565635870870232784, −5.48824638567029495067172540002, −4.65122608998871320836630042836, −3.81984585371111913546467320435, −2.64324822292909103058522985831, −1.88006912207735485627414535969, −1.13130416097942419222782329075, 0.10172987115274498729358782870, 1.69273681355333414626557578227, 2.70560033013505173096896796198, 3.15304439130195209256745111816, 4.09722811884942858310723711959, 5.055207507784781579476249277, 5.96844438804063975327587173147, 6.879579162080442105258580708170, 7.70577228109909797457438077254, 8.51859380386078707084040658907, 9.26776129791445334758472871811, 10.09149838699937179918568286551, 10.75581899114832965784682273148, 11.23271036605563822792388475567, 12.56493976349467833145621911602, 13.2622272199104469644332252528, 14.27482210206618365200139420888, 14.49189675545355076421213060868, 15.29359721176642702441381352093, 15.975649877941642817622267174948, 16.979725398321139428220793636111, 17.48199372514859955058731493965, 18.653426240012907808600751208335, 19.23756024461413278914878742546, 19.61353054183326462362020549766

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