Properties

Label 1-1792-1792.237-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.844 - 0.534i$
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  + (0.881 + 0.471i)3-s + (0.634 − 0.773i)5-s + (0.555 + 0.831i)9-s + (−0.956 − 0.290i)11-s + (−0.773 + 0.634i)13-s + (0.923 − 0.382i)15-s + (−0.923 − 0.382i)17-s + (0.0980 − 0.995i)19-s + (0.980 + 0.195i)23-s + (−0.195 − 0.980i)25-s + (0.0980 + 0.995i)27-s + (0.290 + 0.956i)29-s + (−0.707 + 0.707i)31-s + (−0.707 − 0.707i)33-s + (0.995 − 0.0980i)37-s + ⋯
L(s)  = 1  + (0.881 + 0.471i)3-s + (0.634 − 0.773i)5-s + (0.555 + 0.831i)9-s + (−0.956 − 0.290i)11-s + (−0.773 + 0.634i)13-s + (0.923 − 0.382i)15-s + (−0.923 − 0.382i)17-s + (0.0980 − 0.995i)19-s + (0.980 + 0.195i)23-s + (−0.195 − 0.980i)25-s + (0.0980 + 0.995i)27-s + (0.290 + 0.956i)29-s + (−0.707 + 0.707i)31-s + (−0.707 − 0.707i)33-s + (0.995 − 0.0980i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.844 - 0.534i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.844 - 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2253370045 - 0.7770385525i\)
\(L(\frac12)\) \(\approx\) \(0.2253370045 - 0.7770385525i\)
\(L(1)\) \(\approx\) \(1.253077324 + 0.02439085481i\)
\(L(1)\) \(\approx\) \(1.253077324 + 0.02439085481i\)

Euler product

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

−20.17483324458817117125499457327, −19.67975860933468437857938617017, −18.61197545537001671456336476446, −18.38801066174267558157213305436, −17.56355270224370824888103915281, −16.81934092624925416921106290894, −15.514577620445396541959512372956, −15.03795777928806292552176586928, −14.49091206163640681820984662555, −13.5524509688564098240006346670, −13.04620182477594756840423004801, −12.40811846473126795046598957367, −11.22372290135573611540758103937, −10.3954534353252100407004280362, −9.76602014042900833917414823994, −9.049234252804890114030189430685, −7.84577369601968483111195457236, −7.61828125982414686286846805587, −6.5628056292175140266569598670, −5.89221026969092738009092535683, −4.79412568589646732133931684667, −3.7354059893820207227034319567, −2.611196267438383698688662766487, −2.45763879478050349954273006181, −1.287986120059451141948541209591, 0.11066013284312632806416586653, 1.432462837054013315546335943250, 2.43182042564785798098793449212, 2.9416502196083122237368851677, 4.30346593487677665239972327837, 4.87395180334997337003953033179, 5.525385680770462719067668919159, 6.89195544659527176402360987209, 7.51625565828190771563827399367, 8.68721018542395974380667550838, 9.004716477894717078892597136124, 9.70102031438799317803085415346, 10.59962757816026502195350084892, 11.29018825133697548596849228210, 12.61176817127995727844735595484, 13.070381445648661934811187955834, 13.80724104480114322230456921151, 14.419963087533553705630634107487, 15.366792584398334348393588224743, 16.01639151679134612025905954777, 16.59295803753394074654901178275, 17.542759951254883761156780161357, 18.20181232904683972998152132894, 19.235721844565849654558802093133, 19.77949607103917564639063813446

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