Properties

Label 2-3920-1.1-c1-0-54
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·3-s + 5-s + 2.82·9-s + 4.82·11-s + 2·13-s + 2.41·15-s + 3.65·17-s − 5.65·19-s + 8.41·23-s + 25-s − 0.414·27-s − 2.17·29-s + 4.82·31-s + 11.6·33-s − 5.65·37-s + 4.82·39-s + 0.171·41-s − 12.8·43-s + 2.82·45-s − 0.343·47-s + 8.82·51-s + 5.65·53-s + 4.82·55-s − 13.6·57-s + 4·59-s − 4.65·61-s + 2·65-s + ⋯
L(s)  = 1  + 1.39·3-s + 0.447·5-s + 0.942·9-s + 1.45·11-s + 0.554·13-s + 0.623·15-s + 0.886·17-s − 1.29·19-s + 1.75·23-s + 0.200·25-s − 0.0797·27-s − 0.403·29-s + 0.867·31-s + 2.02·33-s − 0.929·37-s + 0.773·39-s + 0.0267·41-s − 1.96·43-s + 0.421·45-s − 0.0500·47-s + 1.23·51-s + 0.777·53-s + 0.651·55-s − 1.80·57-s + 0.520·59-s − 0.596·61-s + 0.248·65-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: no
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.035112627\)
\(L(\frac12)\) \(\approx\) \(4.035112627\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 3T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 - 8.41T + 23T^{2} \)
29 \( 1 + 2.17T + 29T^{2} \)
31 \( 1 - 4.82T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 - 0.171T + 41T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 + 0.343T + 47T^{2} \)
53 \( 1 - 5.65T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 4.65T + 61T^{2} \)
67 \( 1 + 6.89T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 7.65T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−8.676983557588477763464023393720, −7.955246458890504125783728259675, −6.93360550182823154904053972020, −6.51433707149009689718525862529, −5.49443092038804786944410092711, −4.47464147831033695207427433012, −3.61849784272256894005738011784, −3.08053526335880286086767773576, −2.00366643940069214283007320340, −1.21536549960699095246516684802, 1.21536549960699095246516684802, 2.00366643940069214283007320340, 3.08053526335880286086767773576, 3.61849784272256894005738011784, 4.47464147831033695207427433012, 5.49443092038804786944410092711, 6.51433707149009689718525862529, 6.93360550182823154904053972020, 7.955246458890504125783728259675, 8.676983557588477763464023393720

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