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Properties

Label 2-3920-1.1-c1-0-74
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s + 9-s + 11-s − 3·13-s − 2·15-s − 2·17-s + 5·19-s − 7·23-s + 25-s − 4·27-s − 6·29-s − 4·31-s + 2·33-s − 5·37-s − 6·39-s − 5·41-s − 6·43-s − 45-s + 9·47-s − 4·51-s + 11·53-s − 55-s + 10·57-s − 8·59-s − 12·61-s + 3·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.832·13-s − 0.516·15-s − 0.485·17-s + 1.14·19-s − 1.45·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.348·33-s − 0.821·37-s − 0.960·39-s − 0.780·41-s − 0.914·43-s − 0.149·45-s + 1.31·47-s − 0.560·51-s + 1.51·53-s − 0.134·55-s + 1.32·57-s − 1.04·59-s − 1.53·61-s + 0.372·65-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−8.045631334587433926565604213185, −7.53951714583345799495522820718, −6.92763674196914902075538325597, −5.82114591664624764438741588864, −5.03504102615739121628881564148, −3.96561896156263916289227193388, −3.48381656930142238917164190065, −2.51948102163040585562191366876, −1.71806050912811555430858132847, 0, 1.71806050912811555430858132847, 2.51948102163040585562191366876, 3.48381656930142238917164190065, 3.96561896156263916289227193388, 5.03504102615739121628881564148, 5.82114591664624764438741588864, 6.92763674196914902075538325597, 7.53951714583345799495522820718, 8.045631334587433926565604213185

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