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Properties

Label 2-333270-1.1-c1-0-15
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 − 6·11-s − 4·13-s + 14-s + 16-s + 6·17-s + 20-s − 6·22-s + 25-s − 4·26-s + 28-s + 6·29-s + 6·31-s + 32-s + 6·34-s + 35-s − 6·37-s + 40-s − 6·41-s − 6·43-s − 6·44-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 − 1.80·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.223·20-s − 1.27·22-s + 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.11·29-s + 1.07·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s − 0.986·37-s + 0.158·40-s − 0.937·41-s − 0.914·43-s − 0.904·44-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.035447333\)
\(L(\frac12)\) \(\approx\) \(3.035447333\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−12.55646125628033, −12.24602104079400, −11.70793209912047, −11.48813699658440, −10.56169248183462, −10.30135414356466, −10.06837807133348, −9.707902123554646, −8.778926614980580, −8.242430691277329, −8.100131142762501, −7.294280945200390, −7.166268584562337, −6.521810444826731, −5.779677635398064, −5.432469090175432, −5.147878127481171, −4.648411154739489, −4.167163542922165, −3.274297076286598, −2.797444837493517, −2.651180226213211, −1.804720022886292, −1.301207802061870, −0.3824360677046021, 0.3824360677046021, 1.301207802061870, 1.804720022886292, 2.651180226213211, 2.797444837493517, 3.274297076286598, 4.167163542922165, 4.648411154739489, 5.147878127481171, 5.432469090175432, 5.779677635398064, 6.521810444826731, 7.166268584562337, 7.294280945200390, 8.100131142762501, 8.242430691277329, 8.778926614980580, 9.707902123554646, 10.06837807133348, 10.30135414356466, 10.56169248183462, 11.48813699658440, 11.70793209912047, 12.24602104079400, 12.55646125628033

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