Properties

Label 2-24150-1.1-c1-0-11
Degree $2$
Conductor $24150$
Sign $1$
Analytic cond. $192.838$
Root an. cond. $13.8866$
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 − 7-s + 8-s + 9-s − 4·11-s − 12-s + 2·13-s − 14-s + 16-s + 6·17-s + 18-s + 21-s − 4·22-s − 23-s − 24-s + 2·26-s − 27-s − 28-s − 2·29-s + 8·31-s + 32-s + 4·33-s + 6·34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.218·21-s − 0.852·22-s − 0.208·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.696·33-s + 1.02·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24150\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(192.838\)
Root analytic conductor: \(13.8866\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 24150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.667607369\)
\(L(\frac12)\) \(\approx\) \(2.667607369\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.38573429619643, −15.03647816043219, −14.12326963013648, −13.84650045980068, −13.18932699160481, −12.71811805254518, −12.22538956521446, −11.78942080239357, −11.05645486749525, −10.58633977653075, −10.09706721295478, −9.599915705466935, −8.700949539099691, −7.946937453185279, −7.599159371770751, −6.848481398760855, −6.205782733050732, −5.604841917340833, −5.322948406318520, −4.471187284480998, −3.867549227257785, −3.070296216014777, −2.559186866169870, −1.490148827051980, −0.6178791290467743, 0.6178791290467743, 1.490148827051980, 2.559186866169870, 3.070296216014777, 3.867549227257785, 4.471187284480998, 5.322948406318520, 5.604841917340833, 6.205782733050732, 6.848481398760855, 7.599159371770751, 7.946937453185279, 8.700949539099691, 9.599915705466935, 10.09706721295478, 10.58633977653075, 11.05645486749525, 11.78942080239357, 12.22538956521446, 12.71811805254518, 13.18932699160481, 13.84650045980068, 14.12326963013648, 15.03647816043219, 15.38573429619643

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