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Properties

Label 1-1792-1792.1333-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.624 + 0.780i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.624 + 0.780i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.624 + 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1864352547 + 0.3879773068i\)
\(L(\frac12)\) \(\approx\) \(0.1864352547 + 0.3879773068i\)
\(L(1)\) \(\approx\) \(1.259691517 - 0.2409141340i\)
\(L(1)\) \(\approx\) \(1.259691517 - 0.2409141340i\)

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

−19.920515614050576569093238929076, −18.95352441193745541251733208754, −18.35988784888674722488749528041, −17.715093418406910766832012918437, −16.59664656158500092610198390665, −16.07064671192673480759295924285, −15.249177530941013258608121304946, −14.502036474874423977678909455880, −13.85993288731066528605395521451, −13.34870317352831405937809289263, −12.50774011464782569090753287934, −11.1512764768482315388436430971, −10.57144100868528230764196398948, −10.08166768851881259433959644423, −9.074786465719923928143532323788, −8.523076028888922445486114343824, −7.69267491222247240460181611335, −6.695295300833172588484083019419, −5.757121638472081131474875804012, −5.13195192885370719807036573573, −3.91846809978411956331550864258, −3.18588733432692114940200906473, −2.46381213310887433148942661045, −1.57803363619667679423642560803, −0.055872731051307182099917784047, 1.39997491202419142950387049284, 1.9606776383273186333767278241, 2.64909315563070988925210068290, 4.03614346526890818844765741544, 4.54936444758810168325777963438, 5.88169913638807074620620308639, 6.46820050358168234331996868298, 7.32764409976205401701581137323, 8.25313422606605846475828889907, 8.898966064855014375580534129804, 9.59380080345730754716972494669, 10.2524320046618280101251498182, 11.43702938748518834277645241911, 12.28932160620917364486239147607, 13.103855795674133496542293991309, 13.444837292895498161843819655897, 14.19036263939695206642287924702, 15.15681597602465198236488515292, 15.57995249084351560975198315755, 16.883999167344076124597153651317, 17.34136560483414266902073112950, 18.18345768295630189651513484446, 18.73346188142772313819090606898, 19.69985171527434164773142630962, 20.22581379193782022586390260488

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