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Properties

Label 1-1792-1792.1339-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.280 - 0.959i$
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  + (0.683 + 0.729i)3-s + (0.910 − 0.412i)5-s + (−0.0654 + 0.997i)9-s + (−0.849 + 0.528i)11-s + (−0.995 + 0.0980i)13-s + (0.923 + 0.382i)15-s + (0.130 − 0.991i)17-s + (−0.162 + 0.986i)19-s + (0.751 − 0.659i)23-s + (0.659 − 0.751i)25-s + (−0.773 + 0.634i)27-s + (−0.471 − 0.881i)29-s + (0.965 − 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.352 + 0.935i)37-s + ⋯
L(s)  = 1  + (0.683 + 0.729i)3-s + (0.910 − 0.412i)5-s + (−0.0654 + 0.997i)9-s + (−0.849 + 0.528i)11-s + (−0.995 + 0.0980i)13-s + (0.923 + 0.382i)15-s + (0.130 − 0.991i)17-s + (−0.162 + 0.986i)19-s + (0.751 − 0.659i)23-s + (0.659 − 0.751i)25-s + (−0.773 + 0.634i)27-s + (−0.471 − 0.881i)29-s + (0.965 − 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.352 + 0.935i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.280 - 0.959i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.280 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.411110363 - 1.057809954i\)
\(L(\frac12)\) \(\approx\) \(1.411110363 - 1.057809954i\)
\(L(1)\) \(\approx\) \(1.281827109 + 0.1407908734i\)
\(L(1)\) \(\approx\) \(1.281827109 + 0.1407908734i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.683 + 0.729i)T \)
5 \( 1 + (0.910 - 0.412i)T \)
11 \( 1 + (-0.849 + 0.528i)T \)
13 \( 1 + (-0.995 + 0.0980i)T \)
17 \( 1 + (0.130 - 0.991i)T \)
19 \( 1 + (-0.162 + 0.986i)T \)
23 \( 1 + (0.751 - 0.659i)T \)
29 \( 1 + (-0.471 - 0.881i)T \)
31 \( 1 + (0.965 - 0.258i)T \)
37 \( 1 + (-0.352 + 0.935i)T \)
41 \( 1 + (-0.980 + 0.195i)T \)
43 \( 1 + (-0.290 - 0.956i)T \)
47 \( 1 + (0.608 - 0.793i)T \)
53 \( 1 + (-0.528 - 0.849i)T \)
59 \( 1 + (-0.582 + 0.812i)T \)
61 \( 1 + (-0.227 - 0.973i)T \)
67 \( 1 + (-0.729 + 0.683i)T \)
71 \( 1 + (-0.831 + 0.555i)T \)
73 \( 1 + (-0.442 - 0.896i)T \)
79 \( 1 + (0.991 - 0.130i)T \)
83 \( 1 + (0.634 - 0.773i)T \)
89 \( 1 + (0.946 - 0.321i)T \)
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

−20.07162531059745357011617587649, −19.24675320234346044201373969085, −18.89314255799907232948043780034, −17.85619460597261717335762737098, −17.53350255751848651102422668393, −16.6888438400145980977492664896, −15.42873042316304749267022368692, −14.93380683317903319255902244252, −14.13506183983482883222281775524, −13.47040171958685957091520497480, −12.917572528436180443852559209922, −12.21410045594632209934607972607, −11.029780322355754523287062981151, −10.41115234501574048573112552569, −9.44288544170594348747289916709, −8.877795313935366241635452823726, −7.89914799964275899198965194212, −7.21855386726153892793340509372, −6.45445732944066570745910172105, −5.63787441349545516651599525037, −4.77679255928696170640493083924, −3.32897153036013714110111067992, −2.78465694484979522572556941324, −1.98514890862557282812772860683, −1.05789540431447940905637901292, 0.2634000798176030570825246343, 1.78900819602388131833496903739, 2.45464218608929351327105546288, 3.22018169762536817899797018434, 4.58091683778103030629198909943, 4.9159645182731904409112905007, 5.771500833604189247279804791309, 6.95432440150681530310639877904, 7.81107273295838516636193635240, 8.5942418078503971585883188409, 9.40181794627904097485812486731, 10.13195533812085287004844802561, 10.32861800026065495765601824375, 11.69671787098878847834407453561, 12.503026961698470606592621576520, 13.435312937084676457927251461281, 13.83458993877691729461732572235, 14.81767128626692488677965060327, 15.2554265060070690606352090986, 16.27252970652190740835545088786, 16.84889008902530862149780126620, 17.49360541137469448560397506255, 18.5816299042003261392580054113, 19.050717512457741296608563739013, 20.245567115908852241021185390943

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