Properties

Label 2-333270-1.1-c1-0-95
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: $\chi_{333270} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
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

−12.78405478846078, −12.26643851885756, −12.00635379459908, −11.18653637880113, −11.01901127765227, −10.48210303679492, −10.03922470370427, −9.649788036002718, −9.230632708976059, −8.609028773793293, −8.418678925206805, −7.794081090568873, −7.210729859223456, −6.940341026989294, −6.295128069705755, −5.863705980157145, −5.496777846630398, −4.780088645287646, −4.325291756274545, −3.577703764025350, −3.032230440464877, −2.556500366447672, −2.086884108236215, −1.206118033803937, −0.8614563008088372, 0, 0.8614563008088372, 1.206118033803937, 2.086884108236215, 2.556500366447672, 3.032230440464877, 3.577703764025350, 4.325291756274545, 4.780088645287646, 5.496777846630398, 5.863705980157145, 6.295128069705755, 6.940341026989294, 7.210729859223456, 7.794081090568873, 8.418678925206805, 8.609028773793293, 9.230632708976059, 9.649788036002718, 10.03922470370427, 10.48210303679492, 11.01901127765227, 11.18653637880113, 12.00635379459908, 12.26643851885756, 12.78405478846078

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