Properties

Label 2-14490-1.1-c1-0-52
Degree $2$
Conductor $14490$
Sign $-1$
Analytic cond. $115.703$
Root an. cond. $10.7565$
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 − 4·13-s + 14-s + 16-s + 2·19-s + 20-s − 23-s + 25-s − 4·26-s + 28-s − 6·29-s + 2·31-s + 32-s + 35-s − 10·37-s + 2·38-s + 40-s − 6·41-s − 4·43-s − 46-s − 6·47-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 − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.458·19-s + 0.223·20-s − 0.208·23-s + 1/5·25-s − 0.784·26-s + 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s + 0.169·35-s − 1.64·37-s + 0.324·38-s + 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.147·46-s − 0.875·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14490\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(115.703\)
Root analytic conductor: \(10.7565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14490,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−16.41664104528539, −15.62338067303976, −15.17550517663952, −14.69674704552395, −14.07958455105495, −13.66333859798755, −13.14066157947187, −12.35250893326888, −12.03566816624502, −11.46095845591734, −10.67578009171494, −10.26204514902243, −9.537827778192677, −9.032270035168255, −8.113791208163789, −7.615947600682880, −6.905543871538469, −6.404838409821572, −5.454454814643366, −5.161767834204562, −4.493320682283809, −3.611238935726512, −2.950994222973927, −2.075865807623234, −1.483459387059458, 0, 1.483459387059458, 2.075865807623234, 2.950994222973927, 3.611238935726512, 4.493320682283809, 5.161767834204562, 5.454454814643366, 6.404838409821572, 6.905543871538469, 7.615947600682880, 8.113791208163789, 9.032270035168255, 9.537827778192677, 10.26204514902243, 10.67578009171494, 11.46095845591734, 12.03566816624502, 12.35250893326888, 13.14066157947187, 13.66333859798755, 14.07958455105495, 14.69674704552395, 15.17550517663952, 15.62338067303976, 16.41664104528539

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