Properties

Label 2-3528-1.1-c1-0-45
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 $1$

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

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: Trivial
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 \( 1 - 2 T + p T^{2} \)
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

−8.256281311226866294496813291431, −7.54997069645197216277073706475, −6.58533741844578296970363838022, −5.78343607874588563482284182474, −5.46524989415668552363508752802, −4.33717318093833859660215502627, −3.44124710132985509655416514471, −2.36826211594438163202639427032, −1.66994087876955429718916737640, 0, 1.66994087876955429718916737640, 2.36826211594438163202639427032, 3.44124710132985509655416514471, 4.33717318093833859660215502627, 5.46524989415668552363508752802, 5.78343607874588563482284182474, 6.58533741844578296970363838022, 7.54997069645197216277073706475, 8.256281311226866294496813291431

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