Properties

Label 1-384-384.317-r1-0-0
Degree $1$
Conductor $384$
Sign $0.970 - 0.242i$
Analytic cond. $41.2665$
Root an. cond. $41.2665$
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  + (0.831 + 0.555i)5-s + (0.923 − 0.382i)7-s + (0.980 − 0.195i)11-s + (0.831 − 0.555i)13-s + (0.707 + 0.707i)17-s + (−0.555 − 0.831i)19-s + (0.382 − 0.923i)23-s + (0.382 + 0.923i)25-s + (−0.980 − 0.195i)29-s i·31-s + (0.980 + 0.195i)35-s + (0.555 − 0.831i)37-s + (−0.382 + 0.923i)41-s + (0.195 + 0.980i)43-s + (−0.707 − 0.707i)47-s + ⋯
L(s)  = 1  + (0.831 + 0.555i)5-s + (0.923 − 0.382i)7-s + (0.980 − 0.195i)11-s + (0.831 − 0.555i)13-s + (0.707 + 0.707i)17-s + (−0.555 − 0.831i)19-s + (0.382 − 0.923i)23-s + (0.382 + 0.923i)25-s + (−0.980 − 0.195i)29-s i·31-s + (0.980 + 0.195i)35-s + (0.555 − 0.831i)37-s + (−0.382 + 0.923i)41-s + (0.195 + 0.980i)43-s + (−0.707 − 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(41.2665\)
Root analytic conductor: \(41.2665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (1:\ ),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.018984941 - 0.3723562775i\)
\(L(\frac12)\) \(\approx\) \(3.018984941 - 0.3723562775i\)
\(L(1)\) \(\approx\) \(1.564337932 - 0.04802616805i\)
\(L(1)\) \(\approx\) \(1.564337932 - 0.04802616805i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.831 + 0.555i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (0.980 - 0.195i)T \)
13 \( 1 + (0.831 - 0.555i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (-0.555 - 0.831i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (-0.980 - 0.195i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.555 - 0.831i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (0.195 + 0.980i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.980 + 0.195i)T \)
59 \( 1 + (-0.831 - 0.555i)T \)
61 \( 1 + (-0.195 + 0.980i)T \)
67 \( 1 + (-0.195 + 0.980i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.555 - 0.831i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
97 \( 1 - iT \)
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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

−24.38805773026259789909223608834, −23.60762627610569298024005749195, −22.52871959638807796324963797797, −21.48852670147573846557380542927, −20.97559882537223339964404077243, −20.20355103269163715334085875130, −18.909625672117084539420278563542, −18.17578121270127235662668368671, −17.18775205728419649285551212436, −16.638931034077111859089260353179, −15.43613500615948578362271665448, −14.30013829033246074470031914950, −13.86524021888420550812915025531, −12.59585138583991396808886534107, −11.78935800226380204137661970608, −10.84235840621003497360801377651, −9.55564422594966539008647489688, −8.94184296709729953017416456457, −7.941745702789715917964388995252, −6.61741019699664075871670051175, −5.633815780792042220143745772679, −4.76717565219998735453789672680, −3.547227155335145586956723735432, −1.8731178097289266203234501554, −1.27346789084154716986125887066, 0.989788098823894394137583637493, 2.02838740352038095386375824475, 3.38035152675460641553419580444, 4.50641613482824076490643552586, 5.79525479126679307707531414951, 6.52743727153011191459873587837, 7.73187087083071671738572850173, 8.729304738374371585423454434927, 9.76886306982164887227690464598, 10.86593529835505692097547605784, 11.30862625678072709859740244323, 12.79687889843808116907134143690, 13.59842550734473225730276636949, 14.61816269378010846857198909059, 14.98930109378892821140614125087, 16.61459539715599187160446733865, 17.23589843300467265486024284877, 18.06352282236439422283018320082, 18.86448071768200704238918059546, 19.98532368399441260352479403465, 20.92867455132235071191836637029, 21.570288532975703592858944559692, 22.49618964105551133155060662069, 23.343503993113532336949860693188, 24.38546391687924214375885788111

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