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Properties

Label 2-333270-1.1-c1-0-19
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 − 6·11-s − 4·13-s + 14-s + 16-s + 6·17-s − 8·19-s − 20-s + 6·22-s + 25-s + 4·26-s − 28-s − 6·29-s − 4·31-s − 32-s − 6·34-s + 35-s + 4·37-s + 8·38-s + 40-s − 6·41-s + 10·43-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 − 1.80·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.83·19-s − 0.223·20-s + 1.27·22-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s + 0.657·37-s + 1.29·38-s + 0.158·40-s − 0.937·41-s + 1.52·43-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−12.73985447621162, −12.38885559666968, −12.00624078293392, −11.27034826640797, −10.88268807204675, −10.55064940106195, −10.02995289893660, −9.765047072546842, −9.183786669716992, −8.679330856142060, −8.082827955950532, −7.720419522139800, −7.553659936269182, −6.944618981914022, −6.357444316337482, −5.788336708723955, −5.375599432965772, −4.834393709224551, −4.308005577335362, −3.553105280802765, −3.115343381047681, −2.505532540229247, −2.132187444102715, −1.409517725010842, −0.4059136682840968, 0, 0.4059136682840968, 1.409517725010842, 2.132187444102715, 2.505532540229247, 3.115343381047681, 3.553105280802765, 4.308005577335362, 4.834393709224551, 5.375599432965772, 5.788336708723955, 6.357444316337482, 6.944618981914022, 7.553659936269182, 7.720419522139800, 8.082827955950532, 8.679330856142060, 9.183786669716992, 9.765047072546842, 10.02995289893660, 10.55064940106195, 10.88268807204675, 11.27034826640797, 12.00624078293392, 12.38885559666968, 12.73985447621162

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