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Properties

Label 2-333270-1.1-c1-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 2·11-s + 4·13-s − 14-s + 16-s − 6·17-s + 4·19-s − 20-s − 2·22-s + 25-s + 4·26-s − 28-s − 2·29-s + 2·31-s + 32-s − 6·34-s + 35-s − 4·37-s + 4·38-s − 40-s + 2·41-s − 4·43-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 − 0.603·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.223·20-s − 0.426·22-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 0.371·29-s + 0.359·31-s + 0.176·32-s − 1.02·34-s + 0.169·35-s − 0.657·37-s + 0.648·38-s − 0.158·40-s + 0.312·41-s − 0.609·43-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: \(0\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.540504609\)
\(L(\frac12)\) \(\approx\) \(2.540504609\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.62731832460532, −12.25278992187172, −11.64865402378389, −11.13802011967873, −11.07125194039263, −10.44532874554087, −9.958968923103485, −9.339017340343866, −8.954090282527992, −8.359798956940334, −7.981941782921774, −7.456372206885071, −6.896699452558934, −6.483487492490354, −6.135004672684534, −5.368920131421514, −5.142040105080775, −4.449403083005807, −3.997348090513397, −3.501154181352219, −3.043632532795861, −2.466558053314021, −1.849688342207088, −1.160488471040599, −0.3765563667908768, 0.3765563667908768, 1.160488471040599, 1.849688342207088, 2.466558053314021, 3.043632532795861, 3.501154181352219, 3.997348090513397, 4.449403083005807, 5.142040105080775, 5.368920131421514, 6.135004672684534, 6.483487492490354, 6.896699452558934, 7.456372206885071, 7.981941782921774, 8.359798956940334, 8.954090282527992, 9.339017340343866, 9.958968923103485, 10.44532874554087, 11.07125194039263, 11.13802011967873, 11.64865402378389, 12.25278992187172, 12.62731832460532

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