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Properties

Label 1-392-392.93-r0-0-0
Degree $1$
Conductor $392$
Sign $0.718 - 0.695i$
Analytic cond. $1.82044$
Root an. cond. $1.82044$
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  + (−0.365 − 0.930i)3-s + (0.988 + 0.149i)5-s + (−0.733 + 0.680i)9-s + (0.733 + 0.680i)11-s + (0.222 − 0.974i)13-s + (−0.222 − 0.974i)15-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (0.900 + 0.433i)27-s + (0.900 − 0.433i)29-s + (−0.5 + 0.866i)31-s + (0.365 − 0.930i)33-s + (−0.826 − 0.563i)37-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)3-s + (0.988 + 0.149i)5-s + (−0.733 + 0.680i)9-s + (0.733 + 0.680i)11-s + (0.222 − 0.974i)13-s + (−0.222 − 0.974i)15-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (0.900 + 0.433i)27-s + (0.900 − 0.433i)29-s + (−0.5 + 0.866i)31-s + (0.365 − 0.930i)33-s + (−0.826 − 0.563i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(1.82044\)
Root analytic conductor: \(1.82044\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 392,\ (0:\ ),\ 0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.328873537 - 0.5380012697i\)
\(L(\frac12)\) \(\approx\) \(1.328873537 - 0.5380012697i\)
\(L(1)\) \(\approx\) \(1.130287295 - 0.2900630137i\)
\(L(1)\) \(\approx\) \(1.130287295 - 0.2900630137i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.365 - 0.930i)T \)
5 \( 1 + (0.988 + 0.149i)T \)
11 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (0.988 - 0.149i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
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

−24.46147797379880054864830426878, −23.76020597024452299501326608110, −22.49048732314862611727484767304, −21.84742136953351391341306952414, −21.31524280302048861600674074865, −20.43003315904549525947376211518, −19.402881757054472882195657631259, −18.26117129169810623422492948026, −17.27759400070089848279060841534, −16.70956383512984213214578545608, −15.934946329919193106539682671941, −14.67133999732562378209250272601, −14.104102556134323441193264436972, −13.01514554509642583557824655416, −11.8020987543161433007544289484, −10.970256894022657945639523741484, −10.06530876883413627790210636991, −9.1579667324582457891385598315, −8.56091996068395203699911832479, −6.63674446096697797215682354014, −6.05533043207624316136897075085, −4.94235220948358960382623555795, −4.00631373472130549325376021203, −2.76341433055764485167760439535, −1.25439396999827557481347375955, 1.13524001785541325884151472750, 2.10863520580257975878956680210, 3.31865215917502004897612335390, 5.11414690699949365817187100000, 5.80067438308150779433445899957, 6.83981999481050701443910524868, 7.60006119145024280240840588552, 8.87961286650478459637209529944, 9.892831574640202441590748041765, 10.83201666670222045339285745831, 12.01508784872323551711011459877, 12.63298628994681980830351337523, 13.79830445188065698038542264743, 14.152920118126789283962120411854, 15.5153457484542243300662383466, 16.70102552609591432300305770413, 17.62236973538182016126924992374, 17.95503492617831549874812914171, 18.94625255047087137432589544345, 19.98717551168407852332632374388, 20.75726558455624660518233021806, 21.96779334970178189846118247984, 22.70579675242332953908815533665, 23.31136752424596874487939260203, 24.61655818731122813225162061187

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