Properties

Label 2-3864-1.1-c1-0-21
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 − 0.318·5-s − 7-s + 9-s + 3.71·11-s − 2.66·13-s − 0.318·15-s + 2.21·17-s − 2.34·19-s − 21-s − 23-s − 4.89·25-s + 27-s + 5.41·29-s + 8.21·31-s + 3.71·33-s + 0.318·35-s + 7.28·37-s − 2.66·39-s + 4.77·41-s − 8.59·43-s − 0.318·45-s − 3.63·47-s + 49-s + 2.21·51-s − 0.615·53-s − 1.18·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.142·5-s − 0.377·7-s + 0.333·9-s + 1.11·11-s − 0.739·13-s − 0.0821·15-s + 0.537·17-s − 0.538·19-s − 0.218·21-s − 0.208·23-s − 0.979·25-s + 0.192·27-s + 1.00·29-s + 1.47·31-s + 0.646·33-s + 0.0538·35-s + 1.19·37-s − 0.427·39-s + 0.746·41-s − 1.31·43-s − 0.0474·45-s − 0.529·47-s + 0.142·49-s + 0.310·51-s − 0.0845·53-s − 0.159·55-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

\(L(1)\) \(\approx\) \(2.295854402\)
\(L(\frac12)\) \(\approx\) \(2.295854402\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.318T + 5T^{2} \)
11 \( 1 - 3.71T + 11T^{2} \)
13 \( 1 + 2.66T + 13T^{2} \)
17 \( 1 - 2.21T + 17T^{2} \)
19 \( 1 + 2.34T + 19T^{2} \)
29 \( 1 - 5.41T + 29T^{2} \)
31 \( 1 - 8.21T + 31T^{2} \)
37 \( 1 - 7.28T + 37T^{2} \)
41 \( 1 - 4.77T + 41T^{2} \)
43 \( 1 + 8.59T + 43T^{2} \)
47 \( 1 + 3.63T + 47T^{2} \)
53 \( 1 + 0.615T + 53T^{2} \)
59 \( 1 - 9.81T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 0.747T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 3.84T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + 1.11T + 83T^{2} \)
89 \( 1 - 6.32T + 89T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−8.165688259797286784097197200989, −8.119958304801792136105330311313, −6.83979530637195047236957197549, −6.54393346100189036665163910761, −5.51308456165119964214643463390, −4.47176411660843647095106733514, −3.89123347643662564648022710960, −2.96290861533994160386820347479, −2.10457102055332021738824245190, −0.864054895750017514213802042036, 0.864054895750017514213802042036, 2.10457102055332021738824245190, 2.96290861533994160386820347479, 3.89123347643662564648022710960, 4.47176411660843647095106733514, 5.51308456165119964214643463390, 6.54393346100189036665163910761, 6.83979530637195047236957197549, 8.119958304801792136105330311313, 8.165688259797286784097197200989

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