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Properties

Label 2-333270-1.1-c1-0-122
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 − 3·11-s + 2·13-s + 14-s + 16-s + 3·17-s + 8·19-s − 20-s − 3·22-s + 25-s + 2·26-s + 28-s + 6·29-s − 10·31-s + 32-s + 3·34-s − 35-s + 2·37-s + 8·38-s − 40-s − 6·41-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 − 0.904·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 1.83·19-s − 0.223·20-s − 0.639·22-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 1.11·29-s − 1.79·31-s + 0.176·32-s + 0.514·34-s − 0.169·35-s + 0.328·37-s + 1.29·38-s − 0.158·40-s − 0.937·41-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: 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−12.69066917812758, −12.35309532361429, −12.09463160096997, −11.42642808124043, −10.98089232614691, −10.86670781370357, −10.14859888592583, −9.669518563590781, −9.246870232329234, −8.562971183544125, −8.062085483446901, −7.670755137005266, −7.389299984125553, −6.761063677314630, −6.261709602914167, −5.515537193468963, −5.245254827810663, −5.045726699437397, −4.106475240286996, −3.806630812756157, −3.253778911646176, −2.737238040451739, −2.234553725414484, −1.292452552324270, −1.011447675134692, 0, 1.011447675134692, 1.292452552324270, 2.234553725414484, 2.737238040451739, 3.253778911646176, 3.806630812756157, 4.106475240286996, 5.045726699437397, 5.245254827810663, 5.515537193468963, 6.261709602914167, 6.761063677314630, 7.389299984125553, 7.670755137005266, 8.062085483446901, 8.562971183544125, 9.246870232329234, 9.669518563590781, 10.14859888592583, 10.86670781370357, 10.98089232614691, 11.42642808124043, 12.09463160096997, 12.35309532361429, 12.69066917812758

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