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Properties

Label 1-1792-1792.1131-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.0388 + 0.999i$
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  + (−0.162 + 0.986i)3-s + (−0.973 − 0.227i)5-s + (−0.946 − 0.321i)9-s + (−0.910 + 0.412i)11-s + (−0.956 − 0.290i)13-s + (0.382 − 0.923i)15-s + (−0.608 + 0.793i)17-s + (−0.999 − 0.0327i)19-s + (−0.442 − 0.896i)23-s + (0.896 + 0.442i)25-s + (0.471 − 0.881i)27-s + (0.995 − 0.0980i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.849 + 0.528i)37-s + ⋯
L(s)  = 1  + (−0.162 + 0.986i)3-s + (−0.973 − 0.227i)5-s + (−0.946 − 0.321i)9-s + (−0.910 + 0.412i)11-s + (−0.956 − 0.290i)13-s + (0.382 − 0.923i)15-s + (−0.608 + 0.793i)17-s + (−0.999 − 0.0327i)19-s + (−0.442 − 0.896i)23-s + (0.896 + 0.442i)25-s + (0.471 − 0.881i)27-s + (0.995 − 0.0980i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.849 + 0.528i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0388 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0388 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0388 + 0.999i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.0388 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1147040115 + 0.1103341762i\)
\(L(\frac12)\) \(\approx\) \(0.1147040115 + 0.1103341762i\)
\(L(1)\) \(\approx\) \(0.5342234017 + 0.1040422396i\)
\(L(1)\) \(\approx\) \(0.5342234017 + 0.1040422396i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.162 + 0.986i)T \)
5 \( 1 + (-0.973 - 0.227i)T \)
11 \( 1 + (-0.910 + 0.412i)T \)
13 \( 1 + (-0.956 - 0.290i)T \)
17 \( 1 + (-0.608 + 0.793i)T \)
19 \( 1 + (-0.999 - 0.0327i)T \)
23 \( 1 + (-0.442 - 0.896i)T \)
29 \( 1 + (0.995 - 0.0980i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (-0.849 + 0.528i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (0.773 - 0.634i)T \)
47 \( 1 + (0.130 - 0.991i)T \)
53 \( 1 + (-0.412 - 0.910i)T \)
59 \( 1 + (-0.729 - 0.683i)T \)
61 \( 1 + (0.352 + 0.935i)T \)
67 \( 1 + (-0.986 - 0.162i)T \)
71 \( 1 + (0.195 - 0.980i)T \)
73 \( 1 + (-0.659 - 0.751i)T \)
79 \( 1 + (-0.793 + 0.608i)T \)
83 \( 1 + (-0.881 + 0.471i)T \)
89 \( 1 + (-0.997 + 0.0654i)T \)
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

−19.596194411415197148714925062034, −19.188935595675220756236417642734, −18.45967610824953922721869272356, −17.7325988721925120082842469204, −17.054956900235439449611277432105, −16.00071968227348226876030429665, −15.60074445819949695508203727662, −14.47300412079505505890020924620, −13.94784195867203429105055139649, −12.999697956064483361339938617021, −12.35323209183471486190698203020, −11.704719680280478186416358608474, −11.00991162414252634788261445655, −10.25916118067340011770938791916, −8.97782823026281723886616943624, −8.26900711671519589515671269329, −7.53070245396344321349027665978, −6.99934957226878004509225467481, −6.14770257340374684631756923561, −5.09765788792736286907098465662, −4.37361612967479167463891092657, −3.02928176174643577988810965623, −2.552348246896764431458536854408, −1.337041684110249029737951955482, −0.09472540267449556120942138458, 0.28367117833435403865850695048, 2.1368269143747724105954950896, 2.97966447133112631599743337107, 4.05442014403597834818442825650, 4.55643689351258088958124952636, 5.26039677755979870033253287085, 6.328651435174703035417128589312, 7.28069177741986355685705623967, 8.34625580206156452814395926415, 8.586657409753981007387448558886, 9.863806570523107274060754014105, 10.4351647370566814000680597723, 11.023362829221355256928219551619, 12.09599089784161729065999167169, 12.4589887189796330976571640125, 13.50692812577992451925179401898, 14.63935807498655516983805461642, 15.24618424809588483302041372492, 15.56560012851682212663769051836, 16.48855370269551766823334317734, 17.126965964494441980725547320933, 17.80120510104756962283076525435, 18.93352103808577095421282523667, 19.55618741508892486481057621581, 20.321784877329772264694250890985

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