Properties

Label 2-221067-1.1-c1-0-14
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 − 4·13-s + 14-s − 16-s − 3·17-s − 5·19-s − 2·20-s + 3·23-s − 25-s + 4·26-s + 28-s − 29-s − 9·31-s − 5·32-s + 3·34-s − 2·35-s − 2·37-s + 5·38-s + 6·40-s − 5·41-s − 8·43-s − 3·46-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 − 1.10·13-s + 0.267·14-s − 1/4·16-s − 0.727·17-s − 1.14·19-s − 0.447·20-s + 0.625·23-s − 1/5·25-s + 0.784·26-s + 0.188·28-s − 0.185·29-s − 1.61·31-s − 0.883·32-s + 0.514·34-s − 0.338·35-s − 0.328·37-s + 0.811·38-s + 0.948·40-s − 0.780·41-s − 1.21·43-s − 0.442·46-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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−13.24916615430618, −12.91463280084416, −12.37680423521939, −11.78039894441897, −11.16755004557339, −10.68567107950482, −10.24455034589534, −9.925846049548499, −9.310793905875559, −9.197367806497776, −8.537263245618396, −8.227790338496359, −7.480766931802484, −6.978011263747079, −6.745513981372832, −5.925296084073016, −5.516221686914024, −4.900129704504418, −4.579995870682128, −3.844342152227679, −3.340863992708544, −2.472392604843230, −1.960407005563943, −1.624417117545082, −0.5578303912675606, 0, 0.5578303912675606, 1.624417117545082, 1.960407005563943, 2.472392604843230, 3.340863992708544, 3.844342152227679, 4.579995870682128, 4.900129704504418, 5.516221686914024, 5.925296084073016, 6.745513981372832, 6.978011263747079, 7.480766931802484, 8.227790338496359, 8.537263245618396, 9.197367806497776, 9.310793905875559, 9.925846049548499, 10.24455034589534, 10.68567107950482, 11.16755004557339, 11.78039894441897, 12.37680423521939, 12.91463280084416, 13.24916615430618

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