Properties

Label 2-333270-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 6·13-s + 14-s + 16-s − 6·17-s − 4·19-s + 20-s + 25-s + 6·26-s − 28-s + 6·29-s − 32-s + 6·34-s − 35-s + 10·37-s + 4·38-s − 40-s − 2·41-s + 8·43-s − 8·47-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 − 1.66·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 1.11·29-s − 0.176·32-s + 1.02·34-s − 0.169·35-s + 1.64·37-s + 0.648·38-s − 0.158·40-s − 0.312·41-s + 1.21·43-s − 1.16·47-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: $\chi_{333270} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7981995095\)
\(L(\frac12)\) \(\approx\) \(0.7981995095\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.55889878951566, −12.22198589296502, −11.54668430316475, −11.18931420063681, −10.63293811459556, −10.33651768025294, −9.740710289442992, −9.335335907429005, −9.202975617114807, −8.441949528776232, −8.011959148242389, −7.633339593933075, −6.914221368412415, −6.571273469684131, −6.327673878035574, −5.643777771213277, −4.977334266952394, −4.534319154836901, −4.172133835326589, −3.201690074048980, −2.739030384023964, −2.220514552644999, −1.925781372509612, −0.9574376369112214, −0.2923436961514665, 0.2923436961514665, 0.9574376369112214, 1.925781372509612, 2.220514552644999, 2.739030384023964, 3.201690074048980, 4.172133835326589, 4.534319154836901, 4.977334266952394, 5.643777771213277, 6.327673878035574, 6.571273469684131, 6.914221368412415, 7.633339593933075, 8.011959148242389, 8.441949528776232, 9.202975617114807, 9.335335907429005, 9.740710289442992, 10.33651768025294, 10.63293811459556, 11.18931420063681, 11.54668430316475, 12.22198589296502, 12.55889878951566

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