Properties

Label 1-1792-1792.1355-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.999 - 0.0388i$
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  + (−0.812 + 0.582i)3-s + (0.849 − 0.528i)5-s + (0.321 − 0.946i)9-s + (−0.352 + 0.935i)11-s + (0.471 + 0.881i)13-s + (−0.382 + 0.923i)15-s + (0.608 − 0.793i)17-s + (0.683 + 0.729i)19-s + (0.896 − 0.442i)23-s + (0.442 − 0.896i)25-s + (0.290 + 0.956i)27-s + (0.773 + 0.634i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (0.227 − 0.973i)37-s + ⋯
L(s)  = 1  + (−0.812 + 0.582i)3-s + (0.849 − 0.528i)5-s + (0.321 − 0.946i)9-s + (−0.352 + 0.935i)11-s + (0.471 + 0.881i)13-s + (−0.382 + 0.923i)15-s + (0.608 − 0.793i)17-s + (0.683 + 0.729i)19-s + (0.896 − 0.442i)23-s + (0.442 − 0.896i)25-s + (0.290 + 0.956i)27-s + (0.773 + 0.634i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (0.227 − 0.973i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.999 - 0.0388i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.999 - 0.0388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.304409541 - 0.04474747234i\)
\(L(\frac12)\) \(\approx\) \(2.304409541 - 0.04474747234i\)
\(L(1)\) \(\approx\) \(1.101735026 + 0.09176264257i\)
\(L(1)\) \(\approx\) \(1.101735026 + 0.09176264257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.812 + 0.582i)T \)
5 \( 1 + (0.849 - 0.528i)T \)
11 \( 1 + (-0.352 + 0.935i)T \)
13 \( 1 + (0.471 + 0.881i)T \)
17 \( 1 + (0.608 - 0.793i)T \)
19 \( 1 + (0.683 + 0.729i)T \)
23 \( 1 + (0.896 - 0.442i)T \)
29 \( 1 + (0.773 + 0.634i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (0.227 - 0.973i)T \)
41 \( 1 + (0.555 - 0.831i)T \)
43 \( 1 + (0.0980 - 0.995i)T \)
47 \( 1 + (-0.130 + 0.991i)T \)
53 \( 1 + (-0.935 - 0.352i)T \)
59 \( 1 + (0.999 - 0.0327i)T \)
61 \( 1 + (-0.412 + 0.910i)T \)
67 \( 1 + (-0.582 - 0.812i)T \)
71 \( 1 + (-0.980 - 0.195i)T \)
73 \( 1 + (0.751 - 0.659i)T \)
79 \( 1 + (0.793 - 0.608i)T \)
83 \( 1 + (0.956 + 0.290i)T \)
89 \( 1 + (0.0654 + 0.997i)T \)
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

−19.843137272376003984729091776954, −19.067364323605675465091691747266, −18.49182568513222424270485731886, −17.720124991341890076525353571444, −17.36235525462674166798616237171, −16.449222326263669799983602313123, −15.72891625826699731704806237037, −14.81095092452099291581749982937, −13.81869803357491274604301465451, −13.320464037138018424233288213353, −12.74614258027476009097365340485, −11.66363063179015475247325859029, −10.999395369107166448959384172201, −10.42108581194341152809078803982, −9.67255362551862442740095791022, −8.46668406134518634717043360519, −7.79196284920920880572248257759, −6.80063384408421567378943228861, −6.1424503876915092493821637473, −5.533602859540872519213882908817, −4.830078874697572509026357129285, −3.267658911035758447698002917134, −2.72337482783421586368235077876, −1.39154251000137719732580899559, −0.82277526936859899153253223164, 0.63054193843644029477554870234, 1.48696377386796203697555848038, 2.567578605043192725311286700684, 3.79164960014752522222268399584, 4.700220826605874491295622807635, 5.23064405029461579706960155359, 6.03659675947516644974202105778, 6.82012124367671044732610871806, 7.73101987743253124254315580612, 9.10908553130361066039243633747, 9.37618290974758458969281565682, 10.246260529476025712120237846443, 10.86067660805478017623247289168, 11.95926845008214340717657040493, 12.349289769340571389923000388081, 13.265085012510480482219903784494, 14.14274369589159280909840746069, 14.81502628287307345060441219023, 15.929814345370212187086380576331, 16.31487377867229012830380329121, 17.018700812584247901060027452642, 17.79421426449664908633865678042, 18.269488045687639072571509003623, 19.148131819614583073696729266796, 20.516387387029771968621008004792

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