Properties

Label 1-384-384.149-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: $\chi_{384} (149, \cdot )$
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

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

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