Properties

Label 2-3822-1.1-c1-0-64
Degree $2$
Conductor $3822$
Sign $-1$
Analytic cond. $30.5188$
Root an. cond. $5.52438$
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-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 2.41·11-s + 12-s + 13-s − 15-s + 16-s + 0.414·17-s − 18-s − 3.82·19-s − 20-s − 2.41·22-s − 8.65·23-s − 24-s − 4·25-s − 26-s + 27-s + 2.65·29-s + 30-s + 4.24·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.727·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 0.250·16-s + 0.100·17-s − 0.235·18-s − 0.878·19-s − 0.223·20-s − 0.514·22-s − 1.80·23-s − 0.204·24-s − 0.800·25-s − 0.196·26-s + 0.192·27-s + 0.493·29-s + 0.182·30-s + 0.762·31-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: no
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 \( 1 + T + 5T^{2} \)
11 \( 1 - 2.41T + 11T^{2} \)
17 \( 1 - 0.414T + 17T^{2} \)
19 \( 1 + 3.82T + 19T^{2} \)
23 \( 1 + 8.65T + 23T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 - 4.24T + 31T^{2} \)
37 \( 1 + 3.24T + 37T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + 7.65T + 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 - 9.89T + 59T^{2} \)
61 \( 1 - 5.58T + 61T^{2} \)
67 \( 1 - 1.41T + 67T^{2} \)
71 \( 1 - 5.07T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + 0.585T + 89T^{2} \)
97 \( 1 + 7.31T + 97T^{2} \)
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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

−8.333450475069854668767617808663, −7.67497794974218508017453049677, −6.60103553890156028753093974428, −6.33428776092417073425145528071, −5.05641886092505991118724275315, −4.00666036152517151685042112178, −3.49272227002787046183537912527, −2.29050374623032823957853023007, −1.48791601824780027950030084283, 0, 1.48791601824780027950030084283, 2.29050374623032823957853023007, 3.49272227002787046183537912527, 4.00666036152517151685042112178, 5.05641886092505991118724275315, 6.33428776092417073425145528071, 6.60103553890156028753093974428, 7.67497794974218508017453049677, 8.333450475069854668767617808663

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