Properties

Label 2-3920-1.1-c1-0-70
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·3-s − 5-s − 0.561·9-s + 1.56·11-s − 0.438·13-s − 1.56·15-s + 0.438·17-s − 7.12·19-s − 3.12·23-s + 25-s − 5.56·27-s + 6.68·29-s + 2.43·33-s + 6·37-s − 0.684·39-s − 5.12·41-s − 0.876·43-s + 0.561·45-s − 8.68·47-s + 0.684·51-s − 5.12·53-s − 1.56·55-s − 11.1·57-s − 4·59-s − 15.3·61-s + 0.438·65-s − 10.2·67-s + ⋯
L(s)  = 1  + 0.901·3-s − 0.447·5-s − 0.187·9-s + 0.470·11-s − 0.121·13-s − 0.403·15-s + 0.106·17-s − 1.63·19-s − 0.651·23-s + 0.200·25-s − 1.07·27-s + 1.24·29-s + 0.424·33-s + 0.986·37-s − 0.109·39-s − 0.800·41-s − 0.133·43-s + 0.0837·45-s − 1.26·47-s + 0.0958·51-s − 0.703·53-s − 0.210·55-s − 1.47·57-s − 0.520·59-s − 1.96·61-s + 0.0543·65-s − 1.25·67-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: \(1\)
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 \( 1 - 1.56T + 3T^{2} \)
11 \( 1 - 1.56T + 11T^{2} \)
13 \( 1 + 0.438T + 13T^{2} \)
17 \( 1 - 0.438T + 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
23 \( 1 + 3.12T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
43 \( 1 + 0.876T + 43T^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 + 5.12T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 2.43T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 1.12T + 89T^{2} \)
97 \( 1 + 5.80T + 97T^{2} \)
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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

−8.122152840363497163874171911188, −7.66961860114679651126489267335, −6.54328572484194140102944955130, −6.13108024965993165777572245944, −4.85598705645500092368332686501, −4.19111071326445571508763274720, −3.35545520121437657410584336392, −2.59403383504048912179386416757, −1.62049698385777854730253003062, 0, 1.62049698385777854730253003062, 2.59403383504048912179386416757, 3.35545520121437657410584336392, 4.19111071326445571508763274720, 4.85598705645500092368332686501, 6.13108024965993165777572245944, 6.54328572484194140102944955130, 7.66961860114679651126489267335, 8.122152840363497163874171911188

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