Properties

Label 2-384-1.1-c1-0-7
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·5-s − 2·7-s + 9-s − 4·11-s + 2·13-s − 4·15-s − 2·17-s − 8·19-s − 2·21-s − 4·23-s + 11·25-s + 27-s + 6·31-s − 4·33-s + 8·35-s − 2·37-s + 2·39-s + 6·41-s − 4·45-s − 4·47-s − 3·49-s − 2·51-s + 16·55-s − 8·57-s + 4·59-s + 14·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 1.03·15-s − 0.485·17-s − 1.83·19-s − 0.436·21-s − 0.834·23-s + 11/5·25-s + 0.192·27-s + 1.07·31-s − 0.696·33-s + 1.35·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.596·45-s − 0.583·47-s − 3/7·49-s − 0.280·51-s + 2.15·55-s − 1.05·57-s + 0.520·59-s + 1.79·61-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: \(1\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−10.86660959821522152324936701040, −10.10337746972486500130129788437, −8.649034097305583887898795668373, −8.230854059399453444196153124887, −7.30623361332829545195635303299, −6.27798985919626463740519529757, −4.56007962912554840821971374527, −3.77214318657172177768570145308, −2.65190583901467307643981603895, 0, 2.65190583901467307643981603895, 3.77214318657172177768570145308, 4.56007962912554840821971374527, 6.27798985919626463740519529757, 7.30623361332829545195635303299, 8.230854059399453444196153124887, 8.649034097305583887898795668373, 10.10337746972486500130129788437, 10.86660959821522152324936701040

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