Properties

Label 2-384-1.1-c1-0-1
Degree $2$
Conductor $384$
Sign $1$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s + 9-s − 4·11-s + 6·13-s + 6·17-s − 2·21-s + 4·23-s − 5·25-s − 27-s + 4·29-s + 10·31-s + 4·33-s + 2·37-s − 6·39-s − 2·41-s + 8·43-s − 12·47-s − 3·49-s − 6·51-s − 12·53-s − 4·59-s + 2·61-s + 2·63-s + 4·67-s − 4·69-s − 4·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 1.45·17-s − 0.436·21-s + 0.834·23-s − 25-s − 0.192·27-s + 0.742·29-s + 1.79·31-s + 0.696·33-s + 0.328·37-s − 0.960·39-s − 0.312·41-s + 1.21·43-s − 1.75·47-s − 3/7·49-s − 0.840·51-s − 1.64·53-s − 0.520·59-s + 0.256·61-s + 0.251·63-s + 0.488·67-s − 0.481·69-s − 0.474·71-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{384} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.265414252\)
\(L(\frac12)\) \(\approx\) \(1.265414252\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−11.26026602445558485918915487570, −10.59394573181229341824784268538, −9.710853337608498914769336043468, −8.272861522300808832842531458007, −7.83756825226352882885863464519, −6.40139715881748782453956361829, −5.52928834758807112839354938677, −4.57667813854686016923499181308, −3.12662782741322079682173901720, −1.27109672072212401682622254955, 1.27109672072212401682622254955, 3.12662782741322079682173901720, 4.57667813854686016923499181308, 5.52928834758807112839354938677, 6.40139715881748782453956361829, 7.83756825226352882885863464519, 8.272861522300808832842531458007, 9.710853337608498914769336043468, 10.59394573181229341824784268538, 11.26026602445558485918915487570

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