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 + 3.41·11-s + 1.17·13-s + 16-s − 3.41·17-s + 4.82·19-s + 20-s − 3.41·22-s − 3.65·23-s + 25-s − 1.17·26-s + 5.41·29-s + 7.41·31-s − 32-s + 3.41·34-s − 3.41·37-s − 4.82·38-s − 40-s + 3.65·41-s − 7.07·43-s + 3.41·44-s + 3.65·46-s − 2.58·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.02·11-s + 0.324·13-s + 0.250·16-s − 0.828·17-s + 1.10·19-s + 0.223·20-s − 0.727·22-s − 0.762·23-s + 0.200·25-s − 0.229·26-s + 1.00·29-s + 1.33·31-s − 0.176·32-s + 0.585·34-s − 0.561·37-s − 0.783·38-s − 0.158·40-s + 0.571·41-s − 1.07·43-s + 0.514·44-s + 0.539·46-s − 0.377·47-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

\(L(1)\) \(\approx\) \(1.675356483\)
\(L(\frac12)\) \(\approx\) \(1.675356483\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
17 \( 1 + 3.41T + 17T^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 5.41T + 29T^{2} \)
31 \( 1 - 7.41T + 31T^{2} \)
37 \( 1 + 3.41T + 37T^{2} \)
41 \( 1 - 3.65T + 41T^{2} \)
43 \( 1 + 7.07T + 43T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 - 6.48T + 53T^{2} \)
59 \( 1 - 0.828T + 59T^{2} \)
61 \( 1 + 1.65T + 61T^{2} \)
67 \( 1 - 5.89T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 0.485T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−8.465464335452150337620407229382, −7.76982738325878983611731111009, −6.72180655486050609785172066075, −6.49290243910954680031437989975, −5.56247320938621851542079066865, −4.62895493864919932696958207991, −3.69936007845062307469290264697, −2.74571579433716529157191727067, −1.76104732213059710084928950594, −0.845779668306668128899133781579, 0.845779668306668128899133781579, 1.76104732213059710084928950594, 2.74571579433716529157191727067, 3.69936007845062307469290264697, 4.62895493864919932696958207991, 5.56247320938621851542079066865, 6.49290243910954680031437989975, 6.72180655486050609785172066075, 7.76982738325878983611731111009, 8.465464335452150337620407229382

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