Properties

Label 2-3888-1.1-c1-0-39
Degree $2$
Conductor $3888$
Sign $1$
Analytic cond. $31.0458$
Root an. cond. $5.57187$
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  + 3.46·5-s + 7-s + 3.46·11-s + 5·13-s + 19-s + 6.92·23-s + 6.99·25-s − 3.46·29-s − 5·31-s + 3.46·35-s − 37-s + 3.46·41-s + 43-s + 3.46·47-s − 6·49-s − 10.3·53-s + 11.9·55-s − 3.46·59-s + 2·61-s + 17.3·65-s − 8·67-s − 10.3·71-s + 2·73-s + 3.46·77-s + 79-s − 6.92·83-s + 10.3·89-s + ⋯
L(s)  = 1  + 1.54·5-s + 0.377·7-s + 1.04·11-s + 1.38·13-s + 0.229·19-s + 1.44·23-s + 1.39·25-s − 0.643·29-s − 0.898·31-s + 0.585·35-s − 0.164·37-s + 0.541·41-s + 0.152·43-s + 0.505·47-s − 0.857·49-s − 1.42·53-s + 1.61·55-s − 0.450·59-s + 0.256·61-s + 2.14·65-s − 0.977·67-s − 1.23·71-s + 0.234·73-s + 0.394·77-s + 0.112·79-s − 0.760·83-s + 1.10·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3888\)    =    \(2^{4} \cdot 3^{5}\)
Sign: $1$
Analytic conductor: \(31.0458\)
Root analytic conductor: \(5.57187\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3888,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.302387758\)
\(L(\frac12)\) \(\approx\) \(3.302387758\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 3.46T + 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - T + 79T^{2} \)
83 \( 1 + 6.92T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−8.823148884472770583686001033090, −7.72253562023634107256833095808, −6.81574589779543637368539971185, −6.19601898208036417515526512565, −5.63429678017844926628142228106, −4.83706904053187060669113718097, −3.81573367725641187139774016798, −2.92573440025065869437037121154, −1.72304635608116323224123903823, −1.23579631315049212244451551519, 1.23579631315049212244451551519, 1.72304635608116323224123903823, 2.92573440025065869437037121154, 3.81573367725641187139774016798, 4.83706904053187060669113718097, 5.63429678017844926628142228106, 6.19601898208036417515526512565, 6.81574589779543637368539971185, 7.72253562023634107256833095808, 8.823148884472770583686001033090

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