Properties

Label 2-3822-1.1-c1-0-23
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 2.70·5-s + 6-s + 8-s + 9-s − 2.70·10-s − 0.701·11-s + 12-s − 13-s − 2.70·15-s + 16-s + 2.70·17-s + 18-s + 0.701·19-s − 2.70·20-s − 0.701·22-s + 4.70·23-s + 24-s + 2.29·25-s − 26-s + 27-s + 2.70·29-s − 2.70·30-s + 32-s − 0.701·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.20·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.854·10-s − 0.211·11-s + 0.288·12-s − 0.277·13-s − 0.697·15-s + 0.250·16-s + 0.655·17-s + 0.235·18-s + 0.160·19-s − 0.604·20-s − 0.149·22-s + 0.980·23-s + 0.204·24-s + 0.459·25-s − 0.196·26-s + 0.192·27-s + 0.501·29-s − 0.493·30-s + 0.176·32-s − 0.122·33-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: \(0\)
Selberg data: \((2,\ 3822,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.016144367\)
\(L(\frac12)\) \(\approx\) \(3.016144367\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.70T + 5T^{2} \)
11 \( 1 + 0.701T + 11T^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
19 \( 1 - 0.701T + 19T^{2} \)
23 \( 1 - 4.70T + 23T^{2} \)
29 \( 1 - 2.70T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + 3.40T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 14.8T + 59T^{2} \)
61 \( 1 + 1.29T + 61T^{2} \)
67 \( 1 - 5.40T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 1.29T + 73T^{2} \)
79 \( 1 - 9.40T + 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 - 8.80T + 89T^{2} \)
97 \( 1 - 8.80T + 97T^{2} \)
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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

−8.251352455244099970065662024134, −7.73548167499260295173700254230, −7.14448322871862587834218805482, −6.34529538197202045315364619397, −5.25722507828402728268668980639, −4.62120120212975969738603863326, −3.77339413742182160705513686241, −3.20125577427704920430159034803, −2.31748354193214101954152805758, −0.890237578005539139561295283597, 0.890237578005539139561295283597, 2.31748354193214101954152805758, 3.20125577427704920430159034803, 3.77339413742182160705513686241, 4.62120120212975969738603863326, 5.25722507828402728268668980639, 6.34529538197202045315364619397, 7.14448322871862587834218805482, 7.73548167499260295173700254230, 8.251352455244099970065662024134

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