Properties

Label 2-221067-1.1-c1-0-30
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 + 7-s + 3·8-s + 2·13-s − 14-s − 16-s + 6·17-s + 4·19-s + 8·23-s − 5·25-s − 2·26-s − 28-s − 29-s − 4·31-s − 5·32-s − 6·34-s + 2·37-s − 4·38-s + 2·41-s − 12·43-s − 8·46-s + 49-s + 5·50-s − 2·52-s − 2·53-s + 3·56-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s + 0.554·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 1.66·23-s − 25-s − 0.392·26-s − 0.188·28-s − 0.185·29-s − 0.718·31-s − 0.883·32-s − 1.02·34-s + 0.328·37-s − 0.648·38-s + 0.312·41-s − 1.82·43-s − 1.17·46-s + 1/7·49-s + 0.707·50-s − 0.277·52-s − 0.274·53-s + 0.400·56-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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.26711714116320, −12.80640884358902, −12.25616418770079, −11.67096990148101, −11.23425912766409, −10.92203347579591, −10.20225658125921, −9.842378720747357, −9.553234657207904, −8.912302954045991, −8.573141505064965, −7.972367396656932, −7.709009483251809, −7.149410034828950, −6.712921746825453, −5.840930623693351, −5.386226546800840, −5.098337642939230, −4.465633548568052, −3.668229514751283, −3.503376269657406, −2.727227888516398, −1.857558627814940, −1.246981252182141, −0.9206300173100305, 0, 0.9206300173100305, 1.246981252182141, 1.857558627814940, 2.727227888516398, 3.503376269657406, 3.668229514751283, 4.465633548568052, 5.098337642939230, 5.386226546800840, 5.840930623693351, 6.712921746825453, 7.149410034828950, 7.709009483251809, 7.972367396656932, 8.573141505064965, 8.912302954045991, 9.553234657207904, 9.842378720747357, 10.20225658125921, 10.92203347579591, 11.23425912766409, 11.67096990148101, 12.25616418770079, 12.80640884358902, 13.26711714116320

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