Properties

Label 2-3822-1.1-c1-0-42
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 + 3.80·5-s + 6-s − 8-s + 9-s − 3.80·10-s + 5.28·11-s − 12-s + 13-s − 3.80·15-s + 16-s + 6.15·17-s − 18-s + 5.48·19-s + 3.80·20-s − 5.28·22-s + 6.28·23-s + 24-s + 9.48·25-s − 26-s − 27-s − 3·29-s + 3.80·30-s − 8.48·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.70·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.20·10-s + 1.59·11-s − 0.288·12-s + 0.277·13-s − 0.982·15-s + 0.250·16-s + 1.49·17-s − 0.235·18-s + 1.25·19-s + 0.850·20-s − 1.12·22-s + 1.31·23-s + 0.204·24-s + 1.89·25-s − 0.196·26-s − 0.192·27-s − 0.557·29-s + 0.694·30-s − 1.52·31-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: \(0\)
Selberg data: \((2,\ 3822,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.107693576\)
\(L(\frac12)\) \(\approx\) \(2.107693576\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.80T + 5T^{2} \)
11 \( 1 - 5.28T + 11T^{2} \)
17 \( 1 - 6.15T + 17T^{2} \)
19 \( 1 - 5.48T + 19T^{2} \)
23 \( 1 - 6.28T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 8.48T + 31T^{2} \)
37 \( 1 - 1.32T + 37T^{2} \)
41 \( 1 - 5.32T + 41T^{2} \)
43 \( 1 - 1.61T + 43T^{2} \)
47 \( 1 - 7.09T + 47T^{2} \)
53 \( 1 + 0.354T + 53T^{2} \)
59 \( 1 + 1.67T + 59T^{2} \)
61 \( 1 + 9.77T + 61T^{2} \)
67 \( 1 + 8.44T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 - 1.71T + 73T^{2} \)
79 \( 1 - 2.87T + 79T^{2} \)
83 \( 1 + 7.44T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 3.15T + 97T^{2} \)
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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

−8.940300690387277503044143067840, −7.53394327921972488388018782205, −7.08282080682791959150386543959, −6.11239495778470289193625384940, −5.78623511154678449843113967644, −5.06089045538969268194362679472, −3.73730499300540600116237061583, −2.78981587537140695363190875182, −1.43126271091919644905492186046, −1.20964944454226840211785501760, 1.20964944454226840211785501760, 1.43126271091919644905492186046, 2.78981587537140695363190875182, 3.73730499300540600116237061583, 5.06089045538969268194362679472, 5.78623511154678449843113967644, 6.11239495778470289193625384940, 7.08282080682791959150386543959, 7.53394327921972488388018782205, 8.940300690387277503044143067840

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