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Properties

Label 2-333270-1.1-c1-0-49
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 − 4·11-s − 2·13-s − 14-s + 16-s + 2·17-s − 4·19-s + 20-s + 4·22-s + 25-s + 2·26-s + 28-s + 2·29-s − 32-s − 2·34-s + 35-s − 6·37-s + 4·38-s − 40-s + 6·41-s + 4·43-s − 4·44-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 − 1.20·11-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 + 0.852·22-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 0.371·29-s − 0.176·32-s − 0.342·34-s + 0.169·35-s − 0.986·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s + 0.609·43-s − 0.603·44-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
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

−12.74899886994876, −12.42883253302538, −11.92822274200175, −11.23994875948930, −10.91061686417386, −10.46796124557280, −10.19632653181489, −9.570199822996711, −9.261201157698685, −8.669002334836734, −8.220634120397045, −7.733558693328299, −7.505659437836055, −6.842273678636462, −6.303975686979943, −5.890704919538582, −5.290377766483826, −4.888312734401585, −4.381824176738912, −3.641183968860815, −2.875709939350714, −2.639739451818668, −1.921048800074482, −1.523139926053223, −0.6523225039378676, 0, 0.6523225039378676, 1.523139926053223, 1.921048800074482, 2.639739451818668, 2.875709939350714, 3.641183968860815, 4.381824176738912, 4.888312734401585, 5.290377766483826, 5.890704919538582, 6.303975686979943, 6.842273678636462, 7.505659437836055, 7.733558693328299, 8.220634120397045, 8.669002334836734, 9.261201157698685, 9.570199822996711, 10.19632653181489, 10.46796124557280, 10.91061686417386, 11.23994875948930, 11.92822274200175, 12.42883253302538, 12.74899886994876

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