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Properties

Label 2-333270-1.1-c1-0-18
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 − 2·13-s + 14-s + 16-s + 2·17-s + 4·19-s + 20-s + 25-s + 2·26-s − 28-s − 4·29-s + 2·31-s − 32-s − 2·34-s − 35-s + 10·37-s − 4·38-s − 40-s + 6·41-s − 4·43-s − 10·47-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 − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.742·29-s + 0.359·31-s − 0.176·32-s − 0.342·34-s − 0.169·35-s + 1.64·37-s − 0.648·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s − 1.45·47-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: \(0\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.548282112\)
\(L(\frac12)\) \(\approx\) \(1.548282112\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−12.57836695965708, −12.08440543361701, −11.49955290885359, −11.35578941828933, −10.67287459195820, −10.10669368880770, −9.781774335026665, −9.619001330079792, −8.870690680163378, −8.682115366659844, −7.853429799436433, −7.527838638324797, −7.247178453877455, −6.519680289522350, −6.082340214226033, −5.707096619328178, −5.134412841176697, −4.550431521694470, −3.993946810811280, −3.121167347438588, −2.985783462873957, −2.278799399782003, −1.639625767980166, −1.084526684965355, −0.3950160097432986, 0.3950160097432986, 1.084526684965355, 1.639625767980166, 2.278799399782003, 2.985783462873957, 3.121167347438588, 3.993946810811280, 4.550431521694470, 5.134412841176697, 5.707096619328178, 6.082340214226033, 6.519680289522350, 7.247178453877455, 7.527838638324797, 7.853429799436433, 8.682115366659844, 8.870690680163378, 9.619001330079792, 9.781774335026665, 10.10669368880770, 10.67287459195820, 11.35578941828933, 11.49955290885359, 12.08440543361701, 12.57836695965708

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