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Properties

Label 2-333270-1.1-c1-0-125
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 + 6·13-s + 14-s + 16-s + 2·17-s − 20-s + 4·22-s + 25-s + 6·26-s + 28-s − 2·29-s + 32-s + 2·34-s − 35-s − 10·37-s − 40-s − 6·41-s − 8·43-s + 4·44-s + 12·47-s + 49-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 + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.17·26-s + 0.188·28-s − 0.371·29-s + 0.176·32-s + 0.342·34-s − 0.169·35-s − 1.64·37-s − 0.158·40-s − 0.937·41-s − 1.21·43-s + 0.603·44-s + 1.75·47-s + 1/7·49-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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.71439633631271, −12.31608377256854, −11.88080102817640, −11.65474989111336, −10.98912771080137, −10.73808371554884, −10.35431472961781, −9.473808400363742, −9.205310288796168, −8.594854771543687, −8.215686162895790, −7.804805313898791, −7.070809322745776, −6.735327168937066, −6.314270702254003, −5.752645050940534, −5.299507180247447, −4.765610175195449, −4.060914141081133, −3.844042203616839, −3.345540083154780, −2.888481827092302, −1.864514108121764, −1.504406274037325, −1.027560579684142, 0, 1.027560579684142, 1.504406274037325, 1.864514108121764, 2.888481827092302, 3.345540083154780, 3.844042203616839, 4.060914141081133, 4.765610175195449, 5.299507180247447, 5.752645050940534, 6.314270702254003, 6.735327168937066, 7.070809322745776, 7.804805313898791, 8.215686162895790, 8.594854771543687, 9.205310288796168, 9.473808400363742, 10.35431472961781, 10.73808371554884, 10.98912771080137, 11.65474989111336, 11.88080102817640, 12.31608377256854, 12.71439633631271

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