Properties

Label 2-221067-1.1-c1-0-19
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 + 2·13-s − 14-s − 16-s + 4·17-s − 2·19-s + 2·20-s − 25-s + 2·26-s + 28-s − 29-s − 2·31-s + 5·32-s + 4·34-s + 2·35-s + 2·37-s − 2·38-s + 6·40-s + 10·47-s + 49-s − 50-s − 2·52-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 + 0.554·13-s − 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.458·19-s + 0.447·20-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.185·29-s − 0.359·31-s + 0.883·32-s + 0.685·34-s + 0.338·35-s + 0.328·37-s − 0.324·38-s + 0.948·40-s + 1.45·47-s + 1/7·49-s − 0.141·50-s − 0.277·52-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.01923873029766, −12.75297011315431, −12.43346386355525, −11.91809798127909, −11.40322615228391, −11.08954219792720, −10.38916811865460, −9.919433040363163, −9.446771356874640, −8.925976338533249, −8.492043731475766, −7.971920506266217, −7.580786522887133, −6.970904191395546, −6.365823929492640, −5.859297097995105, −5.511738721440354, −4.857986250672533, −4.305081647528642, −3.871838225078775, −3.504456524945811, −2.986409789861602, −2.331493115275697, −1.399661331499403, −0.6700716939160224, 0, 0.6700716939160224, 1.399661331499403, 2.331493115275697, 2.986409789861602, 3.504456524945811, 3.871838225078775, 4.305081647528642, 4.857986250672533, 5.511738721440354, 5.859297097995105, 6.365823929492640, 6.970904191395546, 7.580786522887133, 7.971920506266217, 8.492043731475766, 8.925976338533249, 9.446771356874640, 9.919433040363163, 10.38916811865460, 11.08954219792720, 11.40322615228391, 11.91809798127909, 12.43346386355525, 12.75297011315431, 13.01923873029766

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