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Properties

Label 1-1792-1792.565-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.585 - 0.810i$
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  + (−0.910 − 0.412i)3-s + (0.0327 + 0.999i)5-s + (0.659 + 0.751i)9-s + (0.986 − 0.162i)11-s + (0.471 + 0.881i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (0.973 − 0.227i)19-s + (0.0654 − 0.997i)23-s + (−0.997 + 0.0654i)25-s + (−0.290 − 0.956i)27-s + (−0.773 − 0.634i)29-s + (−0.965 + 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.729 − 0.683i)37-s + ⋯
L(s)  = 1  + (−0.910 − 0.412i)3-s + (0.0327 + 0.999i)5-s + (0.659 + 0.751i)9-s + (0.986 − 0.162i)11-s + (0.471 + 0.881i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (0.973 − 0.227i)19-s + (0.0654 − 0.997i)23-s + (−0.997 + 0.0654i)25-s + (−0.290 − 0.956i)27-s + (−0.773 − 0.634i)29-s + (−0.965 + 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.729 − 0.683i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.585 - 0.810i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.585 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2654453059 - 0.5193514528i\)
\(L(\frac12)\) \(\approx\) \(0.2654453059 - 0.5193514528i\)
\(L(1)\) \(\approx\) \(0.8157473668 + 0.01072001818i\)
\(L(1)\) \(\approx\) \(0.8157473668 + 0.01072001818i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.910 - 0.412i)T \)
5 \( 1 + (0.0327 + 0.999i)T \)
11 \( 1 + (0.986 - 0.162i)T \)
13 \( 1 + (0.471 + 0.881i)T \)
17 \( 1 + (0.991 + 0.130i)T \)
19 \( 1 + (0.973 - 0.227i)T \)
23 \( 1 + (0.0654 - 0.997i)T \)
29 \( 1 + (-0.773 - 0.634i)T \)
31 \( 1 + (-0.965 + 0.258i)T \)
37 \( 1 + (-0.729 - 0.683i)T \)
41 \( 1 + (-0.555 + 0.831i)T \)
43 \( 1 + (0.0980 - 0.995i)T \)
47 \( 1 + (-0.793 - 0.608i)T \)
53 \( 1 + (-0.162 - 0.986i)T \)
59 \( 1 + (0.528 + 0.849i)T \)
61 \( 1 + (-0.582 - 0.812i)T \)
67 \( 1 + (-0.412 + 0.910i)T \)
71 \( 1 + (0.980 + 0.195i)T \)
73 \( 1 + (0.946 + 0.321i)T \)
79 \( 1 + (-0.130 - 0.991i)T \)
83 \( 1 + (-0.956 - 0.290i)T \)
89 \( 1 + (0.896 + 0.442i)T \)
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

−20.3800938363796912003462855542, −19.746020944943789111474658286576, −18.63260602554771419182260627513, −17.92474550949544942759311289212, −17.16196446880275306884721214117, −16.69349122799798269056154609181, −15.98458910228980835649590257314, −15.34294957834453940408065729278, −14.412144238897561651327400310355, −13.45247810090405330486112527826, −12.62621348161994785974917749768, −12.05692033065415189919651744162, −11.41182538278511482958546728233, −10.54370478438852383267593528262, −9.512729827194596009890729210654, −9.313845292730285160941861758773, −8.079519270412170522926936843401, −7.30766370228249027367874289960, −6.22723232495749352304915356889, −5.420185465908729022032635568724, −5.07725514083839028241822042436, −3.85833890013713078268940555699, −3.38613056353886756356028718870, −1.458431544219066843398380482024, −1.10353808894497111570489357798, 0.13194546486640777849329224083, 1.32118471008169964917910842733, 2.08668716907731918467106644298, 3.38097813062264259221310591978, 4.064701331525553812296694840371, 5.26556403874546798683149332247, 5.99133143271532475009388023743, 6.78993901101217708400502889312, 7.1786998810477170924988606542, 8.21117943170751019473406707665, 9.333518803937801425891916073241, 10.094067383719948662698459823094, 10.92017137881935298172860909199, 11.566909236294991804558397896422, 12.02066585419320925231594382488, 13.029060856790837204725179175189, 13.95446327893529562836041736526, 14.40182954461402140758347330316, 15.33930297509947264456772950120, 16.40324567901023693598737300696, 16.70223397345373631448626371925, 17.679635362442648617656171068085, 18.33606182794901141983151660728, 18.89128330328654893324347633, 19.443558968359841748810607558937

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