Properties

Label 2-3864-1.1-c1-0-2
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 − 3.57·5-s − 7-s + 9-s − 4.81·11-s − 3.91·13-s − 3.57·15-s − 7.21·17-s − 0.335·19-s − 21-s − 23-s + 7.79·25-s + 27-s + 7.11·29-s − 1.21·31-s − 4.81·33-s + 3.57·35-s + 1.55·37-s − 3.91·39-s − 0.0422·41-s + 8.12·43-s − 3.57·45-s + 4.10·47-s + 49-s − 7.21·51-s + 6.35·53-s + 17.2·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.59·5-s − 0.377·7-s + 0.333·9-s − 1.45·11-s − 1.08·13-s − 0.923·15-s − 1.75·17-s − 0.0769·19-s − 0.218·21-s − 0.208·23-s + 1.55·25-s + 0.192·27-s + 1.32·29-s − 0.218·31-s − 0.838·33-s + 0.604·35-s + 0.256·37-s − 0.626·39-s − 0.00659·41-s + 1.23·43-s − 0.533·45-s + 0.598·47-s + 0.142·49-s − 1.01·51-s + 0.872·53-s + 2.32·55-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

\(L(1)\) \(\approx\) \(0.7327450263\)
\(L(\frac12)\) \(\approx\) \(0.7327450263\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.57T + 5T^{2} \)
11 \( 1 + 4.81T + 11T^{2} \)
13 \( 1 + 3.91T + 13T^{2} \)
17 \( 1 + 7.21T + 17T^{2} \)
19 \( 1 + 0.335T + 19T^{2} \)
29 \( 1 - 7.11T + 29T^{2} \)
31 \( 1 + 1.21T + 31T^{2} \)
37 \( 1 - 1.55T + 37T^{2} \)
41 \( 1 + 0.0422T + 41T^{2} \)
43 \( 1 - 8.12T + 43T^{2} \)
47 \( 1 - 4.10T + 47T^{2} \)
53 \( 1 - 6.35T + 53T^{2} \)
59 \( 1 + 2.08T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 1.19T + 67T^{2} \)
71 \( 1 + 6.89T + 71T^{2} \)
73 \( 1 + 2.73T + 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 + 9.17T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−8.432448380245227080952958434313, −7.70930595847695276693328285306, −7.27284139259301401273118504107, −6.53795175024260358699374480256, −5.27197914330156207007707302537, −4.48326714839826520408167253890, −3.97233242341582138660621618445, −2.82363234506714995320436458847, −2.40617814036859171240769685765, −0.44655101126165193144975689653, 0.44655101126165193144975689653, 2.40617814036859171240769685765, 2.82363234506714995320436458847, 3.97233242341582138660621618445, 4.48326714839826520408167253890, 5.27197914330156207007707302537, 6.53795175024260358699374480256, 7.27284139259301401273118504107, 7.70930595847695276693328285306, 8.432448380245227080952958434313

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