Properties

Label 2-3920-1.1-c1-0-76
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  + 2.41·3-s − 5-s + 2.82·9-s − 4.82·11-s − 0.828·13-s − 2.41·15-s + 0.828·17-s − 2.82·19-s + 2.41·23-s + 25-s − 0.414·27-s − 29-s − 6·31-s − 11.6·33-s − 1.99·39-s + 2.17·41-s − 6.41·43-s − 2.82·45-s + 2·47-s + 1.99·51-s − 6.82·53-s + 4.82·55-s − 6.82·57-s − 12.4·59-s + 11.4·61-s + 0.828·65-s − 12.4·67-s + ⋯
L(s)  = 1  + 1.39·3-s − 0.447·5-s + 0.942·9-s − 1.45·11-s − 0.229·13-s − 0.623·15-s + 0.200·17-s − 0.648·19-s + 0.503·23-s + 0.200·25-s − 0.0797·27-s − 0.185·29-s − 1.07·31-s − 2.02·33-s − 0.320·39-s + 0.339·41-s − 0.978·43-s − 0.421·45-s + 0.291·47-s + 0.280·51-s − 0.937·53-s + 0.651·55-s − 0.904·57-s − 1.62·59-s + 1.47·61-s + 0.102·65-s − 1.51·67-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 \( 1 - 2.41T + 3T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 - 0.828T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 2.41T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 2.17T + 41T^{2} \)
43 \( 1 + 6.41T + 43T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + 6.82T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 4.82T + 73T^{2} \)
79 \( 1 + 9.17T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 2.65T + 89T^{2} \)
97 \( 1 + 0.343T + 97T^{2} \)
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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

−8.119928995173777812942781761215, −7.58566790301385592765185910177, −6.97629223454409264378965851240, −5.80010369114703331750718807753, −4.97011694112183452563035874189, −4.11262001685906042033959309059, −3.22173237827856444670520136259, −2.66649123858515447411599559523, −1.74495896908886396167095089638, 0, 1.74495896908886396167095089638, 2.66649123858515447411599559523, 3.22173237827856444670520136259, 4.11262001685906042033959309059, 4.97011694112183452563035874189, 5.80010369114703331750718807753, 6.97629223454409264378965851240, 7.58566790301385592765185910177, 8.119928995173777812942781761215

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