Properties

Label 1-1792-1792.405-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: $\chi_{1792} (405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.0245 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7653101428 + 0.7843280188i\)
\(L(\frac12)\) \(\approx\) \(0.7653101428 + 0.7843280188i\)
\(L(1)\) \(\approx\) \(0.8325174981 - 0.05687068081i\)
\(L(1)\) \(\approx\) \(0.8325174981 - 0.05687068081i\)

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

−19.776158503103030539552577376712, −19.00168493510655298735600041331, −18.15699207128492698422079447250, −17.84936565302606528493128854548, −16.77792541297081271204443202323, −16.019395368042674332187900623755, −15.58517769563827124824353755329, −14.84053832056931542294195103844, −13.84796753355556564712063108457, −13.317736102478633600686629284100, −12.002906231904937983152754130973, −11.3919604602746342989624420332, −10.98978066675150346590174357495, −10.18795114122406159054765848021, −9.403668952063732884941506491464, −8.501396903267414051752125035365, −7.47436581335585561426302366282, −6.67996840472034566795352726960, −5.92337117425522752964490169321, −5.24214766866903646604836511805, −4.18720393626133311590976012204, −3.317634667418018386166198455797, −2.78692835669702517273145707377, −1.04058716856717365541067123272, −0.27316406319554944509497504636, 1.21574575881443813207411507728, 1.35624209003864699518659529282, 2.75488792114239088567796109829, 4.15179135357925432150080800326, 4.66377843984236560612484683719, 5.62826067140126451845608550557, 6.31858158366795978165379308514, 7.178933986685109494863716530015, 8.14368100697395213295987197359, 8.52736940929617157145549056381, 9.81633226753145238648411256679, 10.42159185368596140294052544245, 11.49421102293141225589099514310, 12.007285064668003893084974927941, 12.80554893623365540248760165949, 13.133352269436716123169584307, 14.20021877489819259938120519599, 15.102925063739708918513573263703, 16.190911377177198705958026974085, 16.39703125415422386994246583580, 17.33574699917595144254369026381, 17.91384085049548771260778793255, 18.68143719628905594768717372024, 19.44304125837166232078559602735, 20.13912164518481089495963820319

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