Properties

Label 2-3520-1.1-c1-0-10
Degree $2$
Conductor $3520$
Sign $1$
Analytic cond. $28.1073$
Root an. cond. $5.30163$
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  − 5-s − 3·9-s − 11-s − 2·13-s + 6·17-s − 4·19-s − 4·23-s + 25-s − 6·29-s + 8·31-s + 2·37-s + 2·41-s + 4·43-s + 3·45-s + 12·47-s − 7·49-s + 2·53-s + 55-s + 4·59-s + 10·61-s + 2·65-s − 16·67-s − 8·71-s + 14·73-s − 8·79-s + 9·81-s − 4·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 9-s − 0.301·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.447·45-s + 1.75·47-s − 49-s + 0.274·53-s + 0.134·55-s + 0.520·59-s + 1.28·61-s + 0.248·65-s − 1.95·67-s − 0.949·71-s + 1.63·73-s − 0.900·79-s + 81-s − 0.439·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(28.1073\)
Root analytic conductor: \(5.30163\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.217124127\)
\(L(\frac12)\) \(\approx\) \(1.217124127\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.463053374436302780309182264232, −7.84894108915792323708688752129, −7.29955233914780263654602073668, −6.14840551067462997632986529816, −5.68225732716517030184731642920, −4.75437184609075657275145983342, −3.88233782109363467472314954582, −2.99968262934907114098963322003, −2.16647240934588242926212147183, −0.62652328274679214892628223425, 0.62652328274679214892628223425, 2.16647240934588242926212147183, 2.99968262934907114098963322003, 3.88233782109363467472314954582, 4.75437184609075657275145983342, 5.68225732716517030184731642920, 6.14840551067462997632986529816, 7.29955233914780263654602073668, 7.84894108915792323708688752129, 8.463053374436302780309182264232

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