Properties

Degree $2$
Conductor $3528$
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 + 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 + 0.0995·101-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

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3528} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.895436648\)
\(L(\frac12)\) \(\approx\) \(1.895436648\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.67070293187078, −17.51471197356886, −17.18703807202821, −16.39495247019814, −15.86692972378564, −15.29217775289786, −14.57973686450090, −14.04246696648252, −13.34418642492486, −12.53714148285869, −11.83631049013057, −11.49561335899857, −10.75641961149087, −10.02875530082317, −9.028121948312117, −8.567112049683611, −8.041640273376104, −6.778848791692407, −6.645076937964038, −5.636294555032566, −4.555823366464305, −3.713216715464100, −3.467354541228426, −1.817929017940317, −0.8623124146073099, 0.8623124146073099, 1.817929017940317, 3.467354541228426, 3.713216715464100, 4.555823366464305, 5.636294555032566, 6.645076937964038, 6.778848791692407, 8.041640273376104, 8.567112049683611, 9.028121948312117, 10.02875530082317, 10.75641961149087, 11.49561335899857, 11.83631049013057, 12.53714148285869, 13.34418642492486, 14.04246696648252, 14.57973686450090, 15.29217775289786, 15.86692972378564, 16.39495247019814, 17.18703807202821, 17.51471197356886, 18.67070293187078

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