Properties

Degree $2$
Conductor $3528$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·13-s + 4·17-s + 4·19-s − 4·23-s − 5·25-s − 2·29-s − 8·31-s − 6·37-s + 12·41-s + 4·43-s − 8·47-s − 6·53-s + 12·59-s − 4·61-s − 4·67-s + 12·71-s − 8·73-s − 16·79-s − 4·83-s + 4·89-s − 16·97-s − 8·101-s + 8·103-s − 8·107-s − 14·109-s − 2·113-s + ⋯
L(s)  = 1  − 1.10·13-s + 0.970·17-s + 0.917·19-s − 0.834·23-s − 25-s − 0.371·29-s − 1.43·31-s − 0.986·37-s + 1.87·41-s + 0.609·43-s − 1.16·47-s − 0.824·53-s + 1.56·59-s − 0.512·61-s − 0.488·67-s + 1.42·71-s − 0.936·73-s − 1.80·79-s − 0.439·83-s + 0.423·89-s − 1.62·97-s − 0.796·101-s + 0.788·103-s − 0.773·107-s − 1.34·109-s − 0.188·113-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: $\chi_{3528} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−7.943292858197333536393193375367, −7.60060339775841296378065315494, −6.84676784569375503981366508419, −5.70606535228152180943165965335, −5.38208157474292441837754641952, −4.28119710078031357633531600946, −3.49723241996278064533857546694, −2.52239391682643836376906423915, −1.49987566281652477142815643518, 0, 1.49987566281652477142815643518, 2.52239391682643836376906423915, 3.49723241996278064533857546694, 4.28119710078031357633531600946, 5.38208157474292441837754641952, 5.70606535228152180943165965335, 6.84676784569375503981366508419, 7.60060339775841296378065315494, 7.943292858197333536393193375367

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