Properties

Label 2-160446-1.1-c1-0-9
Degree $2$
Conductor $160446$
Sign $1$
Analytic cond. $1281.16$
Root an. cond. $35.7934$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 3·5-s − 6-s − 4·7-s − 8-s + 9-s − 3·10-s + 12-s + 13-s + 4·14-s + 3·15-s + 16-s − 17-s − 18-s + 8·19-s + 3·20-s − 4·21-s − 3·23-s − 24-s + 4·25-s − 26-s + 27-s − 4·28-s − 6·29-s − 3·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s + 0.277·13-s + 1.06·14-s + 0.774·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.83·19-s + 0.670·20-s − 0.872·21-s − 0.625·23-s − 0.204·24-s + 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160446\)    =    \(2 \cdot 3 \cdot 11^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(1281.16\)
Root analytic conductor: \(35.7934\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 160446,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.216108757\)
\(L(\frac12)\) \(\approx\) \(2.216108757\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
11 \( 1 \)
13 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−13.33598982977717, −12.88997870800108, −12.48091586894015, −11.85938678659134, −11.24049318041758, −10.70448720348744, −10.21513086707283, −9.658891715049217, −9.544476452840628, −9.051463582075212, −8.857657966529119, −7.801539835714539, −7.497858532134346, −7.068022201400957, −6.377584970108485, −5.923314859880685, −5.672176991836842, −4.999481118887134, −3.950210114856627, −3.568431925503934, −2.965433231999784, −2.450618773554911, −1.889000856069288, −1.260679812543412, −0.4691426177375403, 0.4691426177375403, 1.260679812543412, 1.889000856069288, 2.450618773554911, 2.965433231999784, 3.568431925503934, 3.950210114856627, 4.999481118887134, 5.672176991836842, 5.923314859880685, 6.377584970108485, 7.068022201400957, 7.497858532134346, 7.801539835714539, 8.857657966529119, 9.051463582075212, 9.544476452840628, 9.658891715049217, 10.21513086707283, 10.70448720348744, 11.24049318041758, 11.85938678659134, 12.48091586894015, 12.88997870800108, 13.33598982977717

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