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Properties

Label 2-333270-1.1-c1-0-74
Degree $2$
Conductor $333270$
Sign $-1$
Analytic cond. $2661.17$
Root an. cond. $51.5865$
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 − 5-s − 7-s − 8-s + 10-s + 2·13-s + 14-s + 16-s − 2·17-s + 4·19-s − 20-s + 25-s − 2·26-s − 28-s + 10·29-s − 4·31-s − 32-s + 2·34-s + 35-s + 6·37-s − 4·38-s + 40-s − 2·41-s + 8·43-s − 12·47-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s − 0.188·28-s + 1.85·29-s − 0.718·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s + 0.986·37-s − 0.648·38-s + 0.158·40-s − 0.312·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333270\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(2661.17\)
Root analytic conductor: \(51.5865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−12.72234428862633, −12.33416662030113, −11.77484819927332, −11.43942496208714, −10.93406708979674, −10.61916154310661, −9.983774220681450, −9.668187698446134, −9.054690544609872, −8.834067646431962, −8.204270297634794, −7.679179361599753, −7.549612706386454, −6.734745883485108, −6.277203907994156, −6.179467457651050, −5.195205133791104, −4.867757134272973, −4.224083949721191, −3.585304616194496, −3.148545464024572, −2.656946630218874, −1.967671310561270, −1.238514441899963, −0.7574823163388436, 0, 0.7574823163388436, 1.238514441899963, 1.967671310561270, 2.656946630218874, 3.148545464024572, 3.585304616194496, 4.224083949721191, 4.867757134272973, 5.195205133791104, 6.179467457651050, 6.277203907994156, 6.734745883485108, 7.549612706386454, 7.679179361599753, 8.204270297634794, 8.834067646431962, 9.054690544609872, 9.668187698446134, 9.983774220681450, 10.61916154310661, 10.93406708979674, 11.43942496208714, 11.77484819927332, 12.33416662030113, 12.72234428862633

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