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Properties

Label 2-16245-1.1-c1-0-7
Degree $2$
Conductor $16245$
Sign $-1$
Analytic cond. $129.716$
Root an. cond. $11.3893$
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 + 4·7-s − 3·8-s − 10-s − 4·11-s − 2·13-s + 4·14-s − 16-s − 2·17-s + 20-s − 4·22-s + 4·23-s + 25-s − 2·26-s − 4·28-s − 2·29-s + 5·32-s − 2·34-s − 4·35-s + 6·37-s + 3·40-s − 6·41-s + 8·43-s + 4·44-s + 4·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s − 1.06·8-s − 0.316·10-s − 1.20·11-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.755·28-s − 0.371·29-s + 0.883·32-s − 0.342·34-s − 0.676·35-s + 0.986·37-s + 0.474·40-s − 0.937·41-s + 1.21·43-s + 0.603·44-s + 0.589·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16245\)    =    \(3^{2} \cdot 5 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(129.716\)
Root analytic conductor: \(11.3893\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 16245,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.87951060060536, −15.59866160048803, −14.92969482347150, −14.54083670355847, −14.14598147295844, −13.39652643823348, −12.93408284899613, −12.49489214274861, −11.72843046048931, −11.30169928554776, −10.78304187020159, −10.08298947583866, −9.344672989122720, −8.622783562343726, −8.276795980082519, −7.542233015618757, −7.150221255600763, −6.017915336794983, −5.420819903290018, −4.867253326391264, −4.504585766907321, −3.806809866912113, −2.825833181366516, −2.288620776094795, −1.081410149549704, 0, 1.081410149549704, 2.288620776094795, 2.825833181366516, 3.806809866912113, 4.504585766907321, 4.867253326391264, 5.420819903290018, 6.017915336794983, 7.150221255600763, 7.542233015618757, 8.276795980082519, 8.622783562343726, 9.344672989122720, 10.08298947583866, 10.78304187020159, 11.30169928554776, 11.72843046048931, 12.49489214274861, 12.93408284899613, 13.39652643823348, 14.14598147295844, 14.54083670355847, 14.92969482347150, 15.59866160048803, 15.87951060060536

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