Properties

Degree $2$
Conductor $4410$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s + 6·11-s + 4·13-s + 16-s − 2·19-s − 20-s + 6·22-s + 3·23-s + 25-s + 4·26-s + 3·29-s − 8·31-s + 32-s − 4·37-s − 2·38-s − 40-s + 9·41-s − 7·43-s + 6·44-s + 3·46-s + 50-s + 4·52-s + 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.80·11-s + 1.10·13-s + 1/4·16-s − 0.458·19-s − 0.223·20-s + 1.27·22-s + 0.625·23-s + 1/5·25-s + 0.784·26-s + 0.557·29-s − 1.43·31-s + 0.176·32-s − 0.657·37-s − 0.324·38-s − 0.158·40-s + 1.40·41-s − 1.06·43-s + 0.904·44-s + 0.442·46-s + 0.141·50-s + 0.554·52-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4410} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.442659888\)
\(L(\frac12)\) \(\approx\) \(3.442659888\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.438539401876684989258212691369, −7.44935149012820598471125798977, −6.71229899057828670651407729850, −6.26783616614327022024695196390, −5.39696469153382805762506100973, −4.43323482137427833285312312110, −3.80823904381667241835074198882, −3.27761662240906671499306546886, −1.94165700512029329412075874369, −1.00572145580708979387606425651, 1.00572145580708979387606425651, 1.94165700512029329412075874369, 3.27761662240906671499306546886, 3.80823904381667241835074198882, 4.43323482137427833285312312110, 5.39696469153382805762506100973, 6.26783616614327022024695196390, 6.71229899057828670651407729850, 7.44935149012820598471125798977, 8.438539401876684989258212691369

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