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Properties

Label 2-333270-1.1-c1-0-116
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 + 4·13-s + 14-s + 16-s + 7·17-s − 7·19-s − 20-s + 25-s + 4·26-s + 28-s + 2·29-s + 32-s + 7·34-s − 35-s + 6·37-s − 7·38-s − 40-s − 10·41-s + 43-s + 4·47-s + 49-s + 50-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 + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.69·17-s − 1.60·19-s − 0.223·20-s + 1/5·25-s + 0.784·26-s + 0.188·28-s + 0.371·29-s + 0.176·32-s + 1.20·34-s − 0.169·35-s + 0.986·37-s − 1.13·38-s − 0.158·40-s − 1.56·41-s + 0.152·43-s + 0.583·47-s + 1/7·49-s + 0.141·50-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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.68469948944915, −12.38904890762799, −12.07181954628999, −11.45702512095778, −11.00018937951628, −10.77187960149747, −10.23075626950955, −9.703627048434225, −9.153463238098421, −8.496416489761733, −8.068214005021118, −7.949079983004717, −7.214170469806462, −6.618021824252918, −6.298592555148447, −5.781993955295053, −5.248736339233577, −4.777784274203195, −4.178843154495840, −3.809181004371196, −3.299552063538265, −2.783394025228590, −2.077167912249440, −1.428169186125485, −0.9620955223839987, 0, 0.9620955223839987, 1.428169186125485, 2.077167912249440, 2.783394025228590, 3.299552063538265, 3.809181004371196, 4.178843154495840, 4.777784274203195, 5.248736339233577, 5.781993955295053, 6.298592555148447, 6.618021824252918, 7.214170469806462, 7.949079983004717, 8.068214005021118, 8.496416489761733, 9.153463238098421, 9.703627048434225, 10.23075626950955, 10.77187960149747, 11.00018937951628, 11.45702512095778, 12.07181954628999, 12.38904890762799, 12.68469948944915

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