Properties

Label 2-3648-1.1-c1-0-51
Degree $2$
Conductor $3648$
Sign $-1$
Analytic cond. $29.1294$
Root an. cond. $5.39716$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 3·7-s + 9-s − 5·11-s + 2·13-s + 15-s − 17-s + 19-s − 3·21-s − 4·23-s − 4·25-s − 27-s + 6·29-s + 10·31-s + 5·33-s − 3·35-s − 2·39-s − 11·43-s − 45-s − 9·47-s + 2·49-s + 51-s − 10·53-s + 5·55-s − 57-s + 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.554·13-s + 0.258·15-s − 0.242·17-s + 0.229·19-s − 0.654·21-s − 0.834·23-s − 4/5·25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.870·33-s − 0.507·35-s − 0.320·39-s − 1.67·43-s − 0.149·45-s − 1.31·47-s + 2/7·49-s + 0.140·51-s − 1.37·53-s + 0.674·55-s − 0.132·57-s + 0.520·59-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: yes
Arithmetic: yes
Character: $\chi_{3648} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3648,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.137259653188121390855447482027, −7.65627920571478032514336271694, −6.60850791131037206491890289680, −5.90641506250154868855227026945, −4.90222652055451670072407703452, −4.68719487779910370709247985620, −3.52150549884707699550976993427, −2.43726074335480144289137857493, −1.36309445163470127656106423645, 0, 1.36309445163470127656106423645, 2.43726074335480144289137857493, 3.52150549884707699550976993427, 4.68719487779910370709247985620, 4.90222652055451670072407703452, 5.90641506250154868855227026945, 6.60850791131037206491890289680, 7.65627920571478032514336271694, 8.137259653188121390855447482027

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