Properties

Label 2-3864-1.1-c1-0-67
Degree $2$
Conductor $3864$
Sign $-1$
Analytic cond. $30.8541$
Root an. cond. $5.55465$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2.61·5-s − 7-s + 9-s − 3.43·11-s + 0.821·13-s + 2.61·15-s − 5.79·17-s − 7.43·19-s − 21-s + 23-s + 1.82·25-s + 27-s − 5.79·29-s − 3.07·31-s − 3.43·33-s − 2.61·35-s − 0.922·37-s + 0.821·39-s + 0.566·41-s − 11.6·43-s + 2.61·45-s − 2.35·47-s + 49-s − 5.79·51-s + 0.968·53-s − 8.96·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.16·5-s − 0.377·7-s + 0.333·9-s − 1.03·11-s + 0.227·13-s + 0.674·15-s − 1.40·17-s − 1.70·19-s − 0.218·21-s + 0.208·23-s + 0.364·25-s + 0.192·27-s − 1.07·29-s − 0.552·31-s − 0.597·33-s − 0.441·35-s − 0.151·37-s + 0.131·39-s + 0.0884·41-s − 1.78·43-s + 0.389·45-s − 0.343·47-s + 0.142·49-s − 0.810·51-s + 0.132·53-s − 1.20·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(30.8541\)
Root analytic conductor: \(5.55465\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3864,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 - T \)
good5 \( 1 - 2.61T + 5T^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
13 \( 1 - 0.821T + 13T^{2} \)
17 \( 1 + 5.79T + 17T^{2} \)
19 \( 1 + 7.43T + 19T^{2} \)
29 \( 1 + 5.79T + 29T^{2} \)
31 \( 1 + 3.07T + 31T^{2} \)
37 \( 1 + 0.922T + 37T^{2} \)
41 \( 1 - 0.566T + 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + 2.35T + 47T^{2} \)
53 \( 1 - 0.968T + 53T^{2} \)
59 \( 1 - 1.89T + 59T^{2} \)
61 \( 1 - 2.40T + 61T^{2} \)
67 \( 1 + 4.10T + 67T^{2} \)
71 \( 1 - 1.17T + 71T^{2} \)
73 \( 1 - 8.14T + 73T^{2} \)
79 \( 1 + 2.56T + 79T^{2} \)
83 \( 1 + 6.30T + 83T^{2} \)
89 \( 1 + 7.89T + 89T^{2} \)
97 \( 1 + 1.79T + 97T^{2} \)
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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

−8.380036939831698117804452780446, −7.31697270335692530933227410575, −6.59708236807918416555809074184, −5.97513467970114208074076757894, −5.13431762919321298078630388736, −4.28998917873607582524536316708, −3.29336515869630848053944034861, −2.26178818640748139080905645578, −1.89396719079870897524247629410, 0, 1.89396719079870897524247629410, 2.26178818640748139080905645578, 3.29336515869630848053944034861, 4.28998917873607582524536316708, 5.13431762919321298078630388736, 5.97513467970114208074076757894, 6.59708236807918416555809074184, 7.31697270335692530933227410575, 8.380036939831698117804452780446

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