Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 6·11-s + 6·13-s + 2·17-s − 4·19-s + 2·23-s − 25-s + 8·29-s − 4·31-s − 6·37-s − 10·41-s − 4·43-s + 4·47-s − 4·53-s − 12·55-s + 12·59-s + 2·61-s − 12·65-s + 12·67-s + 6·71-s + 2·73-s − 8·79-s − 4·85-s + 14·89-s + 8·95-s + 2·97-s + 18·101-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.80·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 0.417·23-s − 1/5·25-s + 1.48·29-s − 0.718·31-s − 0.986·37-s − 1.56·41-s − 0.609·43-s + 0.583·47-s − 0.549·53-s − 1.61·55-s + 1.56·59-s + 0.256·61-s − 1.48·65-s + 1.46·67-s + 0.712·71-s + 0.234·73-s − 0.900·79-s − 0.433·85-s + 1.48·89-s + 0.820·95-s + 0.203·97-s + 1.79·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3528} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3528,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.895436648\)
\(L(\frac12)\)  \(\approx\)  \(1.895436648\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.567112049683610940726111982348, −8.041640273376104278693893411052, −6.77884879169240667338509658841, −6.64507693796403783052945350367, −5.63629455503256615058718541499, −4.55582336646430451255261527160, −3.71321671546409998599136251041, −3.46735454122842568343533004657, −1.81792901794031719898466771417, −0.862312414607309896583153295404, 0.862312414607309896583153295404, 1.81792901794031719898466771417, 3.46735454122842568343533004657, 3.71321671546409998599136251041, 4.55582336646430451255261527160, 5.63629455503256615058718541499, 6.64507693796403783052945350367, 6.77884879169240667338509658841, 8.041640273376104278693893411052, 8.567112049683610940726111982348

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