Properties

Label 2-3520-1.1-c1-0-16
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 − 2·7-s − 3·9-s − 11-s + 4·13-s − 4·17-s + 25-s + 6·29-s − 2·35-s + 2·37-s + 6·41-s − 2·43-s − 3·45-s − 3·49-s + 10·53-s − 55-s − 12·59-s + 6·61-s + 6·63-s + 4·65-s + 12·67-s + 16·71-s + 4·73-s + 2·77-s − 4·79-s + 9·81-s − 2·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 9-s − 0.301·11-s + 1.10·13-s − 0.970·17-s + 1/5·25-s + 1.11·29-s − 0.338·35-s + 0.328·37-s + 0.937·41-s − 0.304·43-s − 0.447·45-s − 3/7·49-s + 1.37·53-s − 0.134·55-s − 1.56·59-s + 0.768·61-s + 0.755·63-s + 0.496·65-s + 1.46·67-s + 1.89·71-s + 0.468·73-s + 0.227·77-s − 0.450·79-s + 81-s − 0.219·83-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

\(L(1)\) \(\approx\) \(1.553095011\)
\(L(\frac12)\) \(\approx\) \(1.553095011\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−8.597344779719288907602638819473, −8.035940562653855821779418746516, −6.85667269899173093324578324851, −6.30961469790096626723393594297, −5.72971066068134741309412572095, −4.83284777784030043470160695571, −3.79836956390090775269445181604, −2.96374376382014471097320958505, −2.17666388867997363005053217178, −0.71531896275518796286339281459, 0.71531896275518796286339281459, 2.17666388867997363005053217178, 2.96374376382014471097320958505, 3.79836956390090775269445181604, 4.83284777784030043470160695571, 5.72971066068134741309412572095, 6.30961469790096626723393594297, 6.85667269899173093324578324851, 8.035940562653855821779418746516, 8.597344779719288907602638819473

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