Properties

Label 2-384-8.5-c7-0-8
Degree $2$
Conductor $384$
Sign $-0.707 - 0.707i$
Analytic cond. $119.955$
Root an. cond. $10.9524$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 27i·3-s − 344. i·5-s + 1.66e3·7-s − 729·9-s + 2.28e3i·11-s + 1.15e4i·13-s + 9.29e3·15-s − 1.72e4·17-s − 1.45e4i·19-s + 4.50e4i·21-s − 4.08e4·23-s − 4.04e4·25-s − 1.96e4i·27-s + 1.59e5i·29-s − 1.71e5·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.23i·5-s + 1.83·7-s − 0.333·9-s + 0.516i·11-s + 1.45i·13-s + 0.711·15-s − 0.851·17-s − 0.486i·19-s + 1.06i·21-s − 0.700·23-s − 0.517·25-s − 0.192i·27-s + 1.21i·29-s − 1.03·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(119.955\)
Root analytic conductor: \(10.9524\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :7/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.270000448\)
\(L(\frac12)\) \(\approx\) \(1.270000448\)
\(L(\frac{9}{2})\) 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 \( 1 - 27iT \)
good5 \( 1 + 344. iT - 7.81e4T^{2} \)
7 \( 1 - 1.66e3T + 8.23e5T^{2} \)
11 \( 1 - 2.28e3iT - 1.94e7T^{2} \)
13 \( 1 - 1.15e4iT - 6.27e7T^{2} \)
17 \( 1 + 1.72e4T + 4.10e8T^{2} \)
19 \( 1 + 1.45e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.08e4T + 3.40e9T^{2} \)
29 \( 1 - 1.59e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.71e5T + 2.75e10T^{2} \)
37 \( 1 - 9.02e3iT - 9.49e10T^{2} \)
41 \( 1 + 7.35e5T + 1.94e11T^{2} \)
43 \( 1 - 2.80e5iT - 2.71e11T^{2} \)
47 \( 1 + 5.29e5T + 5.06e11T^{2} \)
53 \( 1 - 5.40e5iT - 1.17e12T^{2} \)
59 \( 1 - 7.27e5iT - 2.48e12T^{2} \)
61 \( 1 + 2.21e6iT - 3.14e12T^{2} \)
67 \( 1 - 2.59e6iT - 6.06e12T^{2} \)
71 \( 1 + 5.24e5T + 9.09e12T^{2} \)
73 \( 1 - 1.37e6T + 1.10e13T^{2} \)
79 \( 1 - 6.61e6T + 1.92e13T^{2} \)
83 \( 1 + 6.38e6iT - 2.71e13T^{2} \)
89 \( 1 + 9.60e6T + 4.42e13T^{2} \)
97 \( 1 + 9.87e6T + 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.60600220562858540037652576811, −9.275137284103501731356583097699, −8.805687759330101094139129291724, −7.984377627800964289495468125576, −6.79785150636743511945135127504, −5.19631202689642882923646632471, −4.76539509637654389676692936519, −4.04201053557396057561832850917, −2.02446455628715034609537460342, −1.37777400558786488045057717935, 0.23967245070304361214897837582, 1.62655948264052987946366071035, 2.51664713737584135192155192956, 3.71167811640470043758541590768, 5.12387181280276676567973909334, 6.01974553457459229284392152303, 7.12519426516757439826858499846, 7.980972505984529575266834433039, 8.455666369461534378065982069564, 10.09627784869820821844028364721

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