Properties

Label 2-221067-1.1-c1-0-5
Degree $2$
Conductor $221067$
Sign $1$
Analytic cond. $1765.22$
Root an. cond. $42.0146$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 5-s − 7-s + 3·8-s + 10-s + 5·13-s + 14-s − 16-s − 4·17-s + 4·19-s + 20-s − 6·23-s − 4·25-s − 5·26-s + 28-s + 29-s + 7·31-s − 5·32-s + 4·34-s + 35-s − 10·37-s − 4·38-s − 3·40-s + 9·43-s + 6·46-s − 7·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s + 1.06·8-s + 0.316·10-s + 1.38·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.917·19-s + 0.223·20-s − 1.25·23-s − 4/5·25-s − 0.980·26-s + 0.188·28-s + 0.185·29-s + 1.25·31-s − 0.883·32-s + 0.685·34-s + 0.169·35-s − 1.64·37-s − 0.648·38-s − 0.474·40-s + 1.37·43-s + 0.884·46-s − 1.02·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221067\)    =    \(3^{2} \cdot 7 \cdot 11^{2} \cdot 29\)
Sign: $1$
Analytic conductor: \(1765.22\)
Root analytic conductor: \(42.0146\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{221067} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 221067,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7645774570\)
\(L(\frac12)\) \(\approx\) \(0.7645774570\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
29 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−12.99811688535634, −12.45148470925120, −12.00420185564027, −11.51564441889454, −11.02109892098152, −10.51625544656213, −10.19260618005631, −9.595477762695958, −9.182984014931519, −8.734841601631899, −8.348188615015784, −7.718078302502987, −7.610532951340207, −6.783494066577906, −6.159924166021284, −6.017002235221343, −5.054029601000023, −4.704962658848622, −4.041646834474869, −3.635766537904426, −3.199479822240863, −2.234596217646433, −1.674954676438239, −0.9888196816477087, −0.3310791318092032, 0.3310791318092032, 0.9888196816477087, 1.674954676438239, 2.234596217646433, 3.199479822240863, 3.635766537904426, 4.041646834474869, 4.704962658848622, 5.054029601000023, 6.017002235221343, 6.159924166021284, 6.783494066577906, 7.610532951340207, 7.718078302502987, 8.348188615015784, 8.734841601631899, 9.182984014931519, 9.595477762695958, 10.19260618005631, 10.51625544656213, 11.02109892098152, 11.51564441889454, 12.00420185564027, 12.45148470925120, 12.99811688535634

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