Properties

Label 2-384-24.5-c6-0-64
Degree $2$
Conductor $384$
Sign $0.163 + 0.986i$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.41 + 26.6i)3-s − 127.·5-s + 467.·7-s + (−689. + 235. i)9-s − 554.·11-s + 357. i·13-s + (−561. − 3.38e3i)15-s + 4.49e3i·17-s + 3.24e3i·19-s + (2.06e3 + 1.24e4i)21-s − 1.74e4i·23-s + 511.·25-s + (−9.31e3 − 1.73e4i)27-s − 1.74e4·29-s + 8.03e3·31-s + ⋯
L(s)  = 1  + (0.163 + 0.986i)3-s − 1.01·5-s + 1.36·7-s + (−0.946 + 0.322i)9-s − 0.416·11-s + 0.162i·13-s + (−0.166 − 1.00i)15-s + 0.915i·17-s + 0.472i·19-s + (0.223 + 1.34i)21-s − 1.43i·23-s + 0.0327·25-s + (−0.473 − 0.880i)27-s − 0.714·29-s + 0.269·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.163 + 0.986i$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ 0.163 + 0.986i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.4680344536\)
\(L(\frac12)\) \(\approx\) \(0.4680344536\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-4.41 - 26.6i)T \)
good5 \( 1 + 127.T + 1.56e4T^{2} \)
7 \( 1 - 467.T + 1.17e5T^{2} \)
11 \( 1 + 554.T + 1.77e6T^{2} \)
13 \( 1 - 357. iT - 4.82e6T^{2} \)
17 \( 1 - 4.49e3iT - 2.41e7T^{2} \)
19 \( 1 - 3.24e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.74e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.74e4T + 5.94e8T^{2} \)
31 \( 1 - 8.03e3T + 8.87e8T^{2} \)
37 \( 1 - 7.94e4iT - 2.56e9T^{2} \)
41 \( 1 + 6.18e4iT - 4.75e9T^{2} \)
43 \( 1 + 4.54e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.59e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.90e5T + 2.21e10T^{2} \)
59 \( 1 + 3.06e5T + 4.21e10T^{2} \)
61 \( 1 + 2.51e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.02e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.62e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.49e5T + 1.51e11T^{2} \)
79 \( 1 - 7.18e5T + 2.43e11T^{2} \)
83 \( 1 - 1.38e5T + 3.26e11T^{2} \)
89 \( 1 + 1.22e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.15e6T + 8.32e11T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.39121193339745174710659243819, −9.093330352814439093110535021924, −8.116921302248323433029332905961, −7.84786608935188167140144824387, −6.15968673688626690826032528674, −4.87170678126632180800292745683, −4.33768968591169505772729676521, −3.26831870144606819490372044645, −1.83119211808523495219119848869, −0.11224284858795867195704544089, 1.04272332817705470681669316290, 2.18419831640837436865958123188, 3.44213424905554442978442141685, 4.75408405768505272494027713213, 5.70778513200794818018619402561, 7.23530297779929627687524190274, 7.65516534169441402917656678337, 8.363245433678442498083760772256, 9.397765787990849710599835717184, 11.01497064000909990665749883016

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