Properties

Label 2-3920-1.1-c1-0-32
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
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  + 3-s + 5-s − 2·9-s + 11-s + 5·13-s + 15-s − 17-s − 6·19-s + 4·23-s + 25-s − 5·27-s + 3·29-s + 2·31-s + 33-s + 8·37-s + 5·39-s + 10·41-s + 2·43-s − 2·45-s − 7·47-s − 51-s − 2·53-s + 55-s − 6·57-s + 14·59-s + 8·61-s + 5·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.301·11-s + 1.38·13-s + 0.258·15-s − 0.242·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 0.359·31-s + 0.174·33-s + 1.31·37-s + 0.800·39-s + 1.56·41-s + 0.304·43-s − 0.298·45-s − 1.02·47-s − 0.140·51-s − 0.274·53-s + 0.134·55-s − 0.794·57-s + 1.82·59-s + 1.02·61-s + 0.620·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.605531822\)
\(L(\frac12)\) \(\approx\) \(2.605531822\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−8.450184006249046629693817613296, −8.052260879626194128453533589888, −6.87355980638310647803350201454, −6.23802356173527712653790454770, −5.67843621588496055592912183765, −4.55859758388220869858089920057, −3.80682353586490070672545101304, −2.88294681601796776522847139799, −2.12821923786521836982449245831, −0.925333057584281818568023868813, 0.925333057584281818568023868813, 2.12821923786521836982449245831, 2.88294681601796776522847139799, 3.80682353586490070672545101304, 4.55859758388220869858089920057, 5.67843621588496055592912183765, 6.23802356173527712653790454770, 6.87355980638310647803350201454, 8.052260879626194128453533589888, 8.450184006249046629693817613296

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