Properties

Label 1-392-392.181-r1-0-0
Degree $1$
Conductor $392$
Sign $0.886 + 0.462i$
Analytic cond. $42.1262$
Root an. cond. $42.1262$
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  + (−0.222 − 0.974i)3-s + (−0.222 − 0.974i)5-s + (−0.900 + 0.433i)9-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.623 + 0.781i)17-s + 19-s + (0.623 + 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.623 + 0.781i)27-s + (−0.623 + 0.781i)29-s − 31-s + (0.222 − 0.974i)33-s + (−0.623 + 0.781i)37-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)3-s + (−0.222 − 0.974i)5-s + (−0.900 + 0.433i)9-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.623 + 0.781i)17-s + 19-s + (0.623 + 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.623 + 0.781i)27-s + (−0.623 + 0.781i)29-s − 31-s + (0.222 − 0.974i)33-s + (−0.623 + 0.781i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.886 + 0.462i$
Analytic conductor: \(42.1262\)
Root analytic conductor: \(42.1262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 392,\ (1:\ ),\ 0.886 + 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9594421713 + 0.2352268126i\)
\(L(\frac12)\) \(\approx\) \(0.9594421713 + 0.2352268126i\)
\(L(1)\) \(\approx\) \(0.8192227891 - 0.2492556323i\)
\(L(1)\) \(\approx\) \(0.8192227891 - 0.2492556323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.623 + 0.781i)T \)
41 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 - T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 - T \)
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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

−24.180133397544674919937249013862, −22.89662605466620884945239075057, −22.35277869291174745675571401015, −21.85130389218300009691792873526, −20.749059396773757580229733908477, −19.83792824104564071490049763359, −18.995311728981366400756456300929, −17.945752000832221403845397965819, −17.0292453046007055841528296215, −16.193385454298539085876203041837, −15.294507769163253424594006827351, −14.45838641116324076588181121679, −13.89579566067433263537842088619, −12.186269238879848831954923245390, −11.38127547541039581003466449018, −10.75474814344420258579178474782, −9.593707104782566115867247796813, −9.02377091944252392329122840545, −7.48885693348575581062117989447, −6.61126397927382219269435709056, −5.499268526037553554418054820635, −4.35095212873743764974210552394, −3.44035103829400016605327858048, −2.40086949537648658797855392114, −0.31980282073187007458940789830, 1.04416477219728821421532467908, 1.92862813302308494795095642282, 3.48406086845555408172160652010, 4.85800152873791325887479524119, 5.673796500113628904869729597190, 6.95072521966626076291869809138, 7.673066812891572240444190089911, 8.75310426209607240884300644525, 9.56812663964071976767828154295, 11.06024139005655187065759241844, 11.97712214637709572518865529848, 12.61104409541842302501835795648, 13.36584783584159646586942277197, 14.44914606487676081119133521430, 15.44079805470522439591051346565, 16.717215071763293455869336111519, 17.23620395789335986757933639134, 18.021955275089777274775065775719, 19.25564764985022201795451799544, 19.8683463804051723026760008622, 20.4728660048249191923456264356, 21.94238409153483251882083175558, 22.57286142743528026433217613109, 23.72920600560412970996155990900, 24.231339380589668167561119539708

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