Properties

Degree $2$
Conductor $3600$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·7-s − 6·11-s + 3·13-s − 2·17-s − 19-s + 2·23-s − 6·29-s − 3·31-s + 6·37-s − 4·41-s + 11·43-s + 10·47-s + 18·49-s − 8·53-s − 6·59-s + 3·61-s − 67-s − 12·71-s − 10·73-s + 30·77-s + 8·79-s + 6·83-s + 16·89-s − 15·91-s + 7·97-s + 8·101-s + 4·103-s + ⋯
L(s)  = 1  − 1.88·7-s − 1.80·11-s + 0.832·13-s − 0.485·17-s − 0.229·19-s + 0.417·23-s − 1.11·29-s − 0.538·31-s + 0.986·37-s − 0.624·41-s + 1.67·43-s + 1.45·47-s + 18/7·49-s − 1.09·53-s − 0.781·59-s + 0.384·61-s − 0.122·67-s − 1.42·71-s − 1.17·73-s + 3.41·77-s + 0.900·79-s + 0.658·83-s + 1.69·89-s − 1.57·91-s + 0.710·97-s + 0.796·101-s + 0.394·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8382652391\)
\(L(\frac12)\) \(\approx\) \(0.8382652391\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−8.717018398127818865178442575005, −7.62517098854814982714999943246, −7.19400428990927881598971530117, −6.02669393993336958194327168053, −5.92046788043366802081720734569, −4.74568234665157290177983219323, −3.72070507699883359252831428538, −3.02199328186925399550527504465, −2.25428150864033789500084419507, −0.50211133432685083550760820273, 0.50211133432685083550760820273, 2.25428150864033789500084419507, 3.02199328186925399550527504465, 3.72070507699883359252831428538, 4.74568234665157290177983219323, 5.92046788043366802081720734569, 6.02669393993336958194327168053, 7.19400428990927881598971530117, 7.62517098854814982714999943246, 8.717018398127818865178442575005

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