Properties

Label 2-333270-1.1-c1-0-127
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 + 6·19-s + 20-s + 2·22-s + 25-s − 2·26-s − 28-s + 8·29-s − 10·31-s + 32-s − 35-s + 10·37-s + 6·38-s + 40-s + 10·41-s − 2·43-s + 2·44-s + 2·47-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 + 1.37·19-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 − 1.79·31-s + 0.176·32-s − 0.169·35-s + 1.64·37-s + 0.973·38-s + 0.158·40-s + 1.56·41-s − 0.304·43-s + 0.301·44-s + 0.291·47-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: $\chi_{333270} (1, \cdot )$
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 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.80114451939054, −12.48410176584913, −11.96875439526923, −11.56075140459437, −11.01075881022064, −10.74077568706228, −9.976624704173598, −9.553296630206024, −9.439712101893470, −8.771289308475785, −8.108098677746914, −7.633545441748792, −7.148460267694937, −6.778919436602276, −6.100984245490925, −5.907857181607745, −5.184236070414809, −4.943129684900749, −4.136559462771581, −3.885491316467157, −3.114049269633255, −2.706413637447777, −2.264677367251196, −1.333040288848000, −1.048028680290590, 0, 1.048028680290590, 1.333040288848000, 2.264677367251196, 2.706413637447777, 3.114049269633255, 3.885491316467157, 4.136559462771581, 4.943129684900749, 5.184236070414809, 5.907857181607745, 6.100984245490925, 6.778919436602276, 7.148460267694937, 7.633545441748792, 8.108098677746914, 8.771289308475785, 9.439712101893470, 9.553296630206024, 9.976624704173598, 10.74077568706228, 11.01075881022064, 11.56075140459437, 11.96875439526923, 12.48410176584913, 12.80114451939054

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