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Properties

Label 1-1792-1792.1453-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.480 - 0.877i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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  + (0.999 − 0.0327i)3-s + (0.935 − 0.352i)5-s + (0.997 − 0.0654i)9-s + (−0.973 + 0.227i)11-s + (−0.634 − 0.773i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (−0.582 + 0.812i)19-s + (0.659 − 0.751i)23-s + (0.751 − 0.659i)25-s + (0.995 − 0.0980i)27-s + (0.956 − 0.290i)29-s + (−0.965 − 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.910 − 0.412i)37-s + ⋯
L(s)  = 1  + (0.999 − 0.0327i)3-s + (0.935 − 0.352i)5-s + (0.997 − 0.0654i)9-s + (−0.973 + 0.227i)11-s + (−0.634 − 0.773i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (−0.582 + 0.812i)19-s + (0.659 − 0.751i)23-s + (0.751 − 0.659i)25-s + (0.995 − 0.0980i)27-s + (0.956 − 0.290i)29-s + (−0.965 − 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.910 − 0.412i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.480 - 0.877i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.480 - 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.197047010 - 1.301592312i\)
\(L(\frac12)\) \(\approx\) \(2.197047010 - 1.301592312i\)
\(L(1)\) \(\approx\) \(1.595830531 - 0.3388459140i\)
\(L(1)\) \(\approx\) \(1.595830531 - 0.3388459140i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.999 - 0.0327i)T \)
5 \( 1 + (0.935 - 0.352i)T \)
11 \( 1 + (-0.973 + 0.227i)T \)
13 \( 1 + (-0.634 - 0.773i)T \)
17 \( 1 + (-0.130 - 0.991i)T \)
19 \( 1 + (-0.582 + 0.812i)T \)
23 \( 1 + (0.659 - 0.751i)T \)
29 \( 1 + (0.956 - 0.290i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (-0.910 - 0.412i)T \)
41 \( 1 + (0.195 - 0.980i)T \)
43 \( 1 + (0.471 + 0.881i)T \)
47 \( 1 + (0.608 + 0.793i)T \)
53 \( 1 + (-0.227 - 0.973i)T \)
59 \( 1 + (0.986 + 0.162i)T \)
61 \( 1 + (0.849 - 0.528i)T \)
67 \( 1 + (-0.0327 - 0.999i)T \)
71 \( 1 + (0.555 - 0.831i)T \)
73 \( 1 + (-0.896 - 0.442i)T \)
79 \( 1 + (0.991 + 0.130i)T \)
83 \( 1 + (0.0980 - 0.995i)T \)
89 \( 1 + (0.321 - 0.946i)T \)
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

−20.422203812494117672085600242613, −19.38740218410066962215310740718, −19.08654336644570016669176101437, −18.17168737184210831144788278780, −17.48870906791424025623642332579, −16.69953835675586520957828051356, −15.69697725164130125038508847940, −15.04968172844570131056362860213, −14.38582240467051207685048144611, −13.64857709426457625022249710095, −13.112820798457260598391606300855, −12.414061593322911582306674242914, −11.093072698335823851174128584089, −10.41155344696260246143420504846, −9.768152698093247149261543474581, −8.91514360855532766269583589321, −8.39778650567997030338829552091, −7.20126505909723755640968022524, −6.81571700527780928058422895496, −5.62554403822042911304155079094, −4.83085338129754634367407508681, −3.81975565355623481959316589997, −2.8005902493390103632324133655, −2.26980563161373238091025689719, −1.38056435854489311534938524851, 0.75819859629498469326765426616, 2.11092433023585776704823095582, 2.48699620753660396236687167288, 3.44771179158266710948932012531, 4.69299649810767889012459239922, 5.20616802349473774794032335265, 6.28142655397161597064524061738, 7.26379708184461421497211917073, 7.93946657483665708648845195370, 8.77476581715121736379296982671, 9.44979073357530007102270456871, 10.231633406957319548336912376576, 10.68650225985520525540070798284, 12.23897340031891877059512317816, 12.80740875109531807204537728068, 13.33035648192969682852212775829, 14.25820089155326038917301909352, 14.660229832336948241314694776871, 15.67661591302201988647151685041, 16.23195255491047420368581913550, 17.27753890560168360912875525264, 17.94947234821048061468152913619, 18.60650030676640022333889176096, 19.38002216261780404608828406192, 20.26249787497230196947977054088

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