Properties

Label 2-221067-1.1-c1-0-35
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 5-s + 7-s + 3·8-s − 10-s − 13-s − 14-s − 16-s − 7·17-s − 20-s + 2·23-s − 4·25-s + 26-s − 28-s + 29-s − 5·31-s − 5·32-s + 7·34-s + 35-s − 6·37-s + 3·40-s − 10·41-s + 3·43-s − 2·46-s + 2·47-s + 49-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 − 0.277·13-s − 0.267·14-s − 1/4·16-s − 1.69·17-s − 0.223·20-s + 0.417·23-s − 4/5·25-s + 0.196·26-s − 0.188·28-s + 0.185·29-s − 0.898·31-s − 0.883·32-s + 1.20·34-s + 0.169·35-s − 0.986·37-s + 0.474·40-s − 1.56·41-s + 0.457·43-s − 0.294·46-s + 0.291·47-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: \(2\)
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 \( 1 - T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.55412090483423, −13.04813656452586, −12.57948091516068, −11.94578514567926, −11.52759681802652, −10.85370255889374, −10.66492670294413, −10.08077423488500, −9.689678651605998, −9.031395367223237, −8.877722314092315, −8.415904400056684, −7.855267103808334, −7.251928594893639, −6.940454801041202, −6.361405509315690, −5.598824060724491, −5.228847448964341, −4.745142113197551, −4.058956658035246, −3.846693841177866, −2.827289668727681, −2.280539161237696, −1.649482162858434, −1.237777215275986, 0, 0, 1.237777215275986, 1.649482162858434, 2.280539161237696, 2.827289668727681, 3.846693841177866, 4.058956658035246, 4.745142113197551, 5.228847448964341, 5.598824060724491, 6.361405509315690, 6.940454801041202, 7.251928594893639, 7.855267103808334, 8.415904400056684, 8.877722314092315, 9.031395367223237, 9.689678651605998, 10.08077423488500, 10.66492670294413, 10.85370255889374, 11.52759681802652, 11.94578514567926, 12.57948091516068, 13.04813656452586, 13.55412090483423

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