Properties

Label 2-33150-1.1-c1-0-16
Degree $2$
Conductor $33150$
Sign $1$
Analytic cond. $264.704$
Root an. cond. $16.2697$
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 + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s + 3·11-s + 12-s + 13-s − 2·14-s + 16-s − 17-s − 18-s + 2·19-s + 2·21-s − 3·22-s − 24-s − 26-s + 27-s + 2·28-s − 6·29-s − 4·31-s − 32-s + 3·33-s + 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.436·21-s − 0.639·22-s − 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.522·33-s + 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33150\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(264.704\)
Root analytic conductor: \(16.2697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 33150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.605594046\)
\(L(\frac12)\) \(\approx\) \(2.605594046\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
13 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−14.84233179646202, −14.59328097367259, −14.18065615544873, −13.41537108726885, −13.00142512277432, −12.24562853743961, −11.71331966948326, −11.26614708174387, −10.74891136003049, −10.15012381433276, −9.469669144300448, −9.003143873941292, −8.720833881537850, −7.916438277631627, −7.546061437113742, −6.978295870627811, −6.329922654361043, −5.616617636722468, −4.996852945056612, −4.061450889640150, −3.713043946776161, −2.826864666805875, −2.015626446182528, −1.520366161731250, −0.6807789178533634, 0.6807789178533634, 1.520366161731250, 2.015626446182528, 2.826864666805875, 3.713043946776161, 4.061450889640150, 4.996852945056612, 5.616617636722468, 6.329922654361043, 6.978295870627811, 7.546061437113742, 7.916438277631627, 8.720833881537850, 9.003143873941292, 9.469669144300448, 10.15012381433276, 10.74891136003049, 11.26614708174387, 11.71331966948326, 12.24562853743961, 13.00142512277432, 13.41537108726885, 14.18065615544873, 14.59328097367259, 14.84233179646202

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