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Properties

Label 1-3864-3864.629-r0-0-0
Degree $1$
Conductor $3864$
Sign $0.682 - 0.731i$
Analytic cond. $17.9443$
Root an. cond. $17.9443$
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  + (0.142 − 0.989i)5-s + (−0.959 + 0.281i)11-s + (0.415 − 0.909i)13-s + (0.841 − 0.540i)17-s + (0.841 + 0.540i)19-s + (−0.959 − 0.281i)25-s + (0.841 − 0.540i)29-s + (0.654 + 0.755i)31-s + (0.142 + 0.989i)37-s + (−0.142 + 0.989i)41-s + (0.654 − 0.755i)43-s + 47-s + (0.415 + 0.909i)53-s + (0.142 + 0.989i)55-s + (−0.415 + 0.909i)59-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)5-s + (−0.959 + 0.281i)11-s + (0.415 − 0.909i)13-s + (0.841 − 0.540i)17-s + (0.841 + 0.540i)19-s + (−0.959 − 0.281i)25-s + (0.841 − 0.540i)29-s + (0.654 + 0.755i)31-s + (0.142 + 0.989i)37-s + (−0.142 + 0.989i)41-s + (0.654 − 0.755i)43-s + 47-s + (0.415 + 0.909i)53-s + (0.142 + 0.989i)55-s + (−0.415 + 0.909i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.682 - 0.731i$
Analytic conductor: \(17.9443\)
Root analytic conductor: \(17.9443\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3864,\ (0:\ ),\ 0.682 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.712321418 - 0.7442503342i\)
\(L(\frac12)\) \(\approx\) \(1.712321418 - 0.7442503342i\)
\(L(1)\) \(\approx\) \(1.133138405 - 0.2376092521i\)
\(L(1)\) \(\approx\) \(1.133138405 - 0.2376092521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.142 - 0.989i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (0.841 + 0.540i)T \)
29 \( 1 + (0.841 - 0.540i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (0.142 + 0.989i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−18.68679548095382239460096854740, −18.04655028567146651392316232173, −17.41211475690486044260038205926, −16.53298780368443324809766655082, −15.80982496463827238075136252157, −15.35250874836548632394608457093, −14.312636920907278726659687008849, −14.038830431469098156050983732617, −13.2784191434551616227067886459, −12.43956818326035290066336754929, −11.6196202235065008047389936260, −10.9989426922758626254138149702, −10.39222097817714299210096461353, −9.717755737395099604422962166285, −8.92005187857427735026697359107, −8.005060000873003434150376778757, −7.405466540655072218822608572686, −6.66628647721632113943757477533, −5.91184279696811142816516591849, −5.2744963056407108430780920999, −4.221133791123704704350922342322, −3.43105161288028161534801209683, −2.71392803261022312655942247548, −1.982438866379485022566810959891, −0.82477148334062231753662321301, 0.74668178115811208335183249124, 1.34666469283722484105434917097, 2.573692122568898339821918782228, 3.20316678468995738440072614247, 4.2533606846829775810674568911, 5.07143639802254121538517498769, 5.5060711721427380411178883529, 6.28691479482055035933118245538, 7.478814751220788460868045047465, 7.96642579302920519828136358083, 8.56854438613295477830740966177, 9.51315561625745717612004550311, 10.09762963211829659267745548193, 10.68052084311630573506569710146, 11.857064029941234750089875750860, 12.22610965371158226397949213084, 12.96489668731783630154308763509, 13.66346548879990672787852015609, 14.132856930998096911059217431712, 15.42449894942196057628119790920, 15.614998008584139958083325104635, 16.43829627803348610306877725911, 17.03541001438418626506157942520, 17.85859607937723898563018673710, 18.32707036045249146938114575276

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