Properties

Label 2-3648-1.1-c1-0-23
Degree $2$
Conductor $3648$
Sign $1$
Analytic cond. $29.1294$
Root an. cond. $5.39716$
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 + 3.37·5-s − 3.37·7-s + 9-s − 0.627·11-s + 4·13-s + 3.37·15-s − 5.37·17-s − 19-s − 3.37·21-s − 4.74·23-s + 6.37·25-s + 27-s + 8.74·29-s + 6·31-s − 0.627·33-s − 11.3·35-s + 4·37-s + 4·39-s + 8.74·41-s + 7.37·43-s + 3.37·45-s + 8.11·47-s + 4.37·49-s − 5.37·51-s + 10·53-s − 2.11·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.50·5-s − 1.27·7-s + 0.333·9-s − 0.189·11-s + 1.10·13-s + 0.870·15-s − 1.30·17-s − 0.229·19-s − 0.735·21-s − 0.989·23-s + 1.27·25-s + 0.192·27-s + 1.62·29-s + 1.07·31-s − 0.109·33-s − 1.92·35-s + 0.657·37-s + 0.640·39-s + 1.36·41-s + 1.12·43-s + 0.502·45-s + 1.18·47-s + 0.624·49-s − 0.752·51-s + 1.37·53-s − 0.285·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $1$
Analytic conductor: \(29.1294\)
Root analytic conductor: \(5.39716\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.827018097\)
\(L(\frac12)\) \(\approx\) \(2.827018097\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 + T \)
good5 \( 1 - 3.37T + 5T^{2} \)
7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 + 0.627T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + 5.37T + 17T^{2} \)
23 \( 1 + 4.74T + 23T^{2} \)
29 \( 1 - 8.74T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 8.74T + 41T^{2} \)
43 \( 1 - 7.37T + 43T^{2} \)
47 \( 1 - 8.11T + 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 + 9.37T + 61T^{2} \)
67 \( 1 + 6.74T + 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 - 2.62T + 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 7.48T + 89T^{2} \)
97 \( 1 - 7.25T + 97T^{2} \)
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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

−8.865578168003350043451479332600, −7.904189633228013452235570287454, −6.82119068679677001151666288793, −6.11641038536744316651594406592, −6.01963285774273875714158790029, −4.63428449061917956665970878760, −3.83915487850879924215278055130, −2.69372763591164927612668097333, −2.31083788160521073923030187553, −0.975589588357200883657981308968, 0.975589588357200883657981308968, 2.31083788160521073923030187553, 2.69372763591164927612668097333, 3.83915487850879924215278055130, 4.63428449061917956665970878760, 6.01963285774273875714158790029, 6.11641038536744316651594406592, 6.82119068679677001151666288793, 7.904189633228013452235570287454, 8.865578168003350043451479332600

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