Properties

Degree $2$
Conductor $9408$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.41·5-s + 9-s + 2·11-s + 2.58·13-s + 3.41·15-s + 2.24·17-s − 2.82·19-s − 7.65·23-s + 6.65·25-s + 27-s + 6.82·29-s + 1.17·31-s + 2·33-s + 4·37-s + 2.58·39-s − 6.24·41-s − 5.65·43-s + 3.41·45-s + 2.82·47-s + 2.24·51-s + 2·53-s + 6.82·55-s − 2.82·57-s − 1.17·59-s + 12.2·61-s + 8.82·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.52·5-s + 0.333·9-s + 0.603·11-s + 0.717·13-s + 0.881·15-s + 0.543·17-s − 0.648·19-s − 1.59·23-s + 1.33·25-s + 0.192·27-s + 1.26·29-s + 0.210·31-s + 0.348·33-s + 0.657·37-s + 0.414·39-s − 0.974·41-s − 0.862·43-s + 0.508·45-s + 0.412·47-s + 0.314·51-s + 0.274·53-s + 0.920·55-s − 0.374·57-s − 0.152·59-s + 1.56·61-s + 1.09·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{9408} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.155459858\)
\(L(\frac12)\) \(\approx\) \(4.155459858\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good5 \( 1 - 3.41T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 7.65T + 23T^{2} \)
29 \( 1 - 6.82T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 1.17T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 9.31T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 7.31T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 2.58T + 97T^{2} \)
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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

−7.927424926919604154160952321092, −6.72203030971948567910003152864, −6.42369736977622451905066485189, −5.77118004697093099328645255204, −5.01363930293116518144775743042, −4.10859726306704098474389389594, −3.41317056138765672993485839396, −2.42107459796500435031814111383, −1.86501110355699153164315223581, −1.00525718599102715798861008895, 1.00525718599102715798861008895, 1.86501110355699153164315223581, 2.42107459796500435031814111383, 3.41317056138765672993485839396, 4.10859726306704098474389389594, 5.01363930293116518144775743042, 5.77118004697093099328645255204, 6.42369736977622451905066485189, 6.72203030971948567910003152864, 7.927424926919604154160952321092

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