Properties

Label 2-3864-1.1-c1-0-39
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2.57·5-s − 7-s + 9-s + 4.81·11-s + 4.91·13-s + 2.57·15-s − 7.21·17-s + 2.33·19-s − 21-s − 23-s + 1.64·25-s + 27-s + 3.10·29-s − 1.21·31-s + 4.81·33-s − 2.57·35-s + 0.223·37-s + 4.91·39-s + 8.25·41-s + 7.31·43-s + 2.57·45-s + 8.11·47-s + 49-s − 7.21·51-s − 1.13·53-s + 12.4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.15·5-s − 0.377·7-s + 0.333·9-s + 1.45·11-s + 1.36·13-s + 0.665·15-s − 1.75·17-s + 0.535·19-s − 0.218·21-s − 0.208·23-s + 0.328·25-s + 0.192·27-s + 0.576·29-s − 0.218·31-s + 0.838·33-s − 0.435·35-s + 0.0368·37-s + 0.786·39-s + 1.28·41-s + 1.11·43-s + 0.384·45-s + 1.18·47-s + 0.142·49-s − 1.01·51-s − 0.156·53-s + 1.67·55-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: \(0\)
Selberg data: \((2,\ 3864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.349636769\)
\(L(\frac12)\) \(\approx\) \(3.349636769\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.57T + 5T^{2} \)
11 \( 1 - 4.81T + 11T^{2} \)
13 \( 1 - 4.91T + 13T^{2} \)
17 \( 1 + 7.21T + 17T^{2} \)
19 \( 1 - 2.33T + 19T^{2} \)
29 \( 1 - 3.10T + 29T^{2} \)
31 \( 1 + 1.21T + 31T^{2} \)
37 \( 1 - 0.223T + 37T^{2} \)
41 \( 1 - 8.25T + 41T^{2} \)
43 \( 1 - 7.31T + 43T^{2} \)
47 \( 1 - 8.11T + 47T^{2} \)
53 \( 1 + 1.13T + 53T^{2} \)
59 \( 1 + 9.56T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 - 6.01T + 67T^{2} \)
71 \( 1 + 4.75T + 71T^{2} \)
73 \( 1 + 9.70T + 73T^{2} \)
79 \( 1 - 5.87T + 79T^{2} \)
83 \( 1 - 4.60T + 83T^{2} \)
89 \( 1 - 7.95T + 89T^{2} \)
97 \( 1 + 4.43T + 97T^{2} \)
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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

−8.913242268091785903696583548090, −7.78633536580710606587864082665, −6.82508292158714920922040925687, −6.24803984425845743423395593296, −5.81772792426996349962759484274, −4.46892459113199247742064672869, −3.90531205389734045700574054712, −2.88707682673903017447448911103, −1.96739873218145017615550098252, −1.12891854608539919332817774087, 1.12891854608539919332817774087, 1.96739873218145017615550098252, 2.88707682673903017447448911103, 3.90531205389734045700574054712, 4.46892459113199247742064672869, 5.81772792426996349962759484274, 6.24803984425845743423395593296, 6.82508292158714920922040925687, 7.78633536580710606587864082665, 8.913242268091785903696583548090

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