Properties

Label 2-3528-1.1-c1-0-2
Degree $2$
Conductor $3528$
Sign $1$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 2·17-s + 2·19-s − 8·23-s + 11·25-s − 2·29-s − 4·31-s − 6·37-s − 2·41-s + 8·43-s − 4·47-s + 10·53-s + 6·59-s − 4·61-s − 12·67-s + 14·73-s − 8·79-s + 6·83-s + 8·85-s + 10·89-s − 8·95-s + 2·97-s + 12·101-s + 12·103-s + 12·107-s + 10·109-s − 6·113-s + ⋯
L(s)  = 1  − 1.78·5-s − 0.485·17-s + 0.458·19-s − 1.66·23-s + 11/5·25-s − 0.371·29-s − 0.718·31-s − 0.986·37-s − 0.312·41-s + 1.21·43-s − 0.583·47-s + 1.37·53-s + 0.781·59-s − 0.512·61-s − 1.46·67-s + 1.63·73-s − 0.900·79-s + 0.658·83-s + 0.867·85-s + 1.05·89-s − 0.820·95-s + 0.203·97-s + 1.19·101-s + 1.18·103-s + 1.16·107-s + 0.957·109-s − 0.564·113-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: $1$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3528} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8274989252\)
\(L(\frac12)\) \(\approx\) \(0.8274989252\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−8.515074793505794327617365568851, −7.69639740408856738387093436756, −7.36849069017085576948583776260, −6.46556088998900741811578342958, −5.49102837407369040609817160877, −4.54986168178647908951568994162, −3.89113776702465302721269609787, −3.28822043116364534811243543739, −2.03505353231784433825131603728, −0.51874703726271079374177335007, 0.51874703726271079374177335007, 2.03505353231784433825131603728, 3.28822043116364534811243543739, 3.89113776702465302721269609787, 4.54986168178647908951568994162, 5.49102837407369040609817160877, 6.46556088998900741811578342958, 7.36849069017085576948583776260, 7.69639740408856738387093436756, 8.515074793505794327617365568851

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