Properties

Label 2-3822-1.1-c1-0-71
Degree $2$
Conductor $3822$
Sign $-1$
Analytic cond. $30.5188$
Root an. cond. $5.52438$
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-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 15-s + 16-s − 5·17-s − 18-s + 19-s + 20-s − 22-s − 5·23-s − 24-s − 4·25-s + 26-s + 27-s − 3·29-s − 30-s + 2·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.213·22-s − 1.04·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.557·29-s − 0.182·30-s + 0.359·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3822\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(30.5188\)
Root analytic conductor: \(5.52438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3822,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 - T \)
7 \( 1 \)
13 \( 1 + T \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.229624320832690142984378383684, −7.52816032351613775475960337785, −6.75482774895766807647838084195, −6.13224739380455376431007248026, −5.15918727573138497040371551761, −4.16478786226631774399464294051, −3.28562902398430010868422011747, −2.21183611182006506064580779862, −1.64319459172741036695391272168, 0, 1.64319459172741036695391272168, 2.21183611182006506064580779862, 3.28562902398430010868422011747, 4.16478786226631774399464294051, 5.15918727573138497040371551761, 6.13224739380455376431007248026, 6.75482774895766807647838084195, 7.52816032351613775475960337785, 8.229624320832690142984378383684

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