Properties

Label 2-3920-1.1-c1-0-73
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  + 2.18·3-s − 5-s + 1.76·9-s − 3.59·11-s − 5.85·13-s − 2.18·15-s + 6.36·17-s + 4.50·19-s − 2.62·23-s + 25-s − 2.68·27-s + 2.05·29-s − 6.62·31-s − 7.85·33-s − 5.91·37-s − 12.7·39-s − 7.22·41-s + 5.34·43-s − 1.76·45-s + 2.31·47-s + 13.8·51-s − 4.10·53-s + 3.59·55-s + 9.83·57-s − 5.07·59-s + 7.37·61-s + 5.85·65-s + ⋯
L(s)  = 1  + 1.26·3-s − 0.447·5-s + 0.589·9-s − 1.08·11-s − 1.62·13-s − 0.563·15-s + 1.54·17-s + 1.03·19-s − 0.548·23-s + 0.200·25-s − 0.517·27-s + 0.382·29-s − 1.19·31-s − 1.36·33-s − 0.972·37-s − 2.04·39-s − 1.12·41-s + 0.815·43-s − 0.263·45-s + 0.338·47-s + 1.94·51-s − 0.564·53-s + 0.485·55-s + 1.30·57-s − 0.660·59-s + 0.944·61-s + 0.726·65-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: \(1\)
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 \( 1 - 2.18T + 3T^{2} \)
11 \( 1 + 3.59T + 11T^{2} \)
13 \( 1 + 5.85T + 13T^{2} \)
17 \( 1 - 6.36T + 17T^{2} \)
19 \( 1 - 4.50T + 19T^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
31 \( 1 + 6.62T + 31T^{2} \)
37 \( 1 + 5.91T + 37T^{2} \)
41 \( 1 + 7.22T + 41T^{2} \)
43 \( 1 - 5.34T + 43T^{2} \)
47 \( 1 - 2.31T + 47T^{2} \)
53 \( 1 + 4.10T + 53T^{2} \)
59 \( 1 + 5.07T + 59T^{2} \)
61 \( 1 - 7.37T + 61T^{2} \)
67 \( 1 + 3.39T + 67T^{2} \)
71 \( 1 + 16.7T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 - 3.25T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−7.936069764460107174260538603550, −7.59043098908619314515062957504, −7.09465936133020941353797205857, −5.60327907018635049002478990945, −5.16169244680729056450784446785, −4.09449635494322505409505776729, −3.13876668177928411668777564603, −2.77663185061918619625674330703, −1.67707080723812019486224651713, 0, 1.67707080723812019486224651713, 2.77663185061918619625674330703, 3.13876668177928411668777564603, 4.09449635494322505409505776729, 5.16169244680729056450784446785, 5.60327907018635049002478990945, 7.09465936133020941353797205857, 7.59043098908619314515062957504, 7.936069764460107174260538603550

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