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Properties

Label 2-333270-1.1-c1-0-106
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 + 2·11-s + 2·13-s − 14-s + 16-s + 2·17-s + 4·19-s − 20-s + 2·22-s + 25-s + 2·26-s − 28-s − 8·29-s + 2·31-s + 32-s + 2·34-s + 35-s − 4·37-s + 4·38-s − 40-s − 2·41-s + 8·43-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 + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.223·20-s + 0.426·22-s + 1/5·25-s + 0.392·26-s − 0.188·28-s − 1.48·29-s + 0.359·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.657·37-s + 0.648·38-s − 0.158·40-s − 0.312·41-s + 1.21·43-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−12.91330696871030, −12.34292765355670, −11.86748604095842, −11.57272128757463, −11.22147867340486, −10.58975517815690, −10.19670074172548, −9.562550907007748, −9.300754939792933, −8.602001668184337, −8.233172188894859, −7.591079015666150, −7.215226551450158, −6.775761777053280, −6.262889981491065, −5.691061354605661, −5.363379396867722, −4.802900662950637, −4.103119309804958, −3.682138654023333, −3.440986614125717, −2.764901701240423, −2.139839862854277, −1.396911783447388, −0.9189424243732772, 0, 0.9189424243732772, 1.396911783447388, 2.139839862854277, 2.764901701240423, 3.440986614125717, 3.682138654023333, 4.103119309804958, 4.802900662950637, 5.363379396867722, 5.691061354605661, 6.262889981491065, 6.775761777053280, 7.215226551450158, 7.591079015666150, 8.233172188894859, 8.602001668184337, 9.300754939792933, 9.562550907007748, 10.19670074172548, 10.58975517815690, 11.22147867340486, 11.57272128757463, 11.86748604095842, 12.34292765355670, 12.91330696871030

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