Properties

Label 2-221067-1.1-c1-0-25
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s + 7-s + 3·8-s − 2·10-s − 6·13-s − 14-s − 16-s − 2·17-s + 8·19-s − 2·20-s − 25-s + 6·26-s − 28-s + 29-s − 4·31-s − 5·32-s + 2·34-s + 2·35-s − 2·37-s − 8·38-s + 6·40-s − 2·41-s + 4·43-s + 4·47-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 0.632·10-s − 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 1.83·19-s − 0.447·20-s − 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.185·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s − 0.328·37-s − 1.29·38-s + 0.948·40-s − 0.312·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-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: \(1\)
Selberg data: \((2,\ 221067,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
29 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.43754039815085, −12.71432682492920, −12.15510378534604, −11.96134094487084, −11.15277229280781, −10.78641420979759, −10.17283907200216, −9.859289097138061, −9.387916041807305, −9.222246175176723, −8.600529778961388, −7.970586181785944, −7.528177833275729, −7.226975307728450, −6.679355963133525, −5.830269416308457, −5.397894625820574, −5.047069348124960, −4.567052608561691, −3.917332422932934, −3.250721779743950, −2.472348113441248, −2.073193005321116, −1.408327579408744, −0.7736571688183792, 0, 0.7736571688183792, 1.408327579408744, 2.073193005321116, 2.472348113441248, 3.250721779743950, 3.917332422932934, 4.567052608561691, 5.047069348124960, 5.397894625820574, 5.830269416308457, 6.679355963133525, 7.226975307728450, 7.528177833275729, 7.970586181785944, 8.600529778961388, 9.222246175176723, 9.387916041807305, 9.859289097138061, 10.17283907200216, 10.78641420979759, 11.15277229280781, 11.96134094487084, 12.15510378534604, 12.71432682492920, 13.43754039815085

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