Properties

Label 2-3920-1.1-c1-0-67
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  + 0.414·3-s + 5-s − 2.82·9-s + 0.828·11-s − 4.82·13-s + 0.414·15-s + 4.82·17-s − 2.82·19-s − 0.414·23-s + 25-s − 2.41·27-s − 29-s + 6·31-s + 0.343·33-s − 1.99·39-s − 7.82·41-s − 3.58·43-s − 2.82·45-s − 2·47-s + 1.99·51-s − 1.17·53-s + 0.828·55-s − 1.17·57-s − 4.48·59-s + 5.48·61-s − 4.82·65-s − 9.58·67-s + ⋯
L(s)  = 1  + 0.239·3-s + 0.447·5-s − 0.942·9-s + 0.249·11-s − 1.33·13-s + 0.106·15-s + 1.17·17-s − 0.648·19-s − 0.0863·23-s + 0.200·25-s − 0.464·27-s − 0.185·29-s + 1.07·31-s + 0.0597·33-s − 0.320·39-s − 1.22·41-s − 0.546·43-s − 0.421·45-s − 0.291·47-s + 0.280·51-s − 0.160·53-s + 0.111·55-s − 0.155·57-s − 0.583·59-s + 0.702·61-s − 0.598·65-s − 1.17·67-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 \( 1 - 0.414T + 3T^{2} \)
11 \( 1 - 0.828T + 11T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 - 4.82T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 0.414T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 + 3.58T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + 1.17T + 53T^{2} \)
59 \( 1 + 4.48T + 59T^{2} \)
61 \( 1 - 5.48T + 61T^{2} \)
67 \( 1 + 9.58T + 67T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 + 0.828T + 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + 8.65T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−8.206011324173688267594849797163, −7.40708693989609610217642267455, −6.62641534382088283647335788525, −5.81033451226306419630331086382, −5.18661549011192502982190408329, −4.33786214732182446865009859525, −3.19633947561842943398030785236, −2.59942774280718036414860003182, −1.54042571092675288000290094150, 0, 1.54042571092675288000290094150, 2.59942774280718036414860003182, 3.19633947561842943398030785236, 4.33786214732182446865009859525, 5.18661549011192502982190408329, 5.81033451226306419630331086382, 6.62641534382088283647335788525, 7.40708693989609610217642267455, 8.206011324173688267594849797163

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