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Properties

Label 2-53067-1.1-c1-0-9
Degree $2$
Conductor $53067$
Sign $1$
Analytic cond. $423.742$
Root an. cond. $20.5849$
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 + 2·5-s + 6-s − 3·8-s + 9-s + 2·10-s + 4·11-s − 12-s − 2·13-s + 2·15-s − 16-s + 6·17-s + 18-s − 2·20-s + 4·22-s − 3·24-s − 25-s − 2·26-s + 27-s + 2·29-s + 2·30-s + 5·32-s + 4·33-s + 6·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.516·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.447·20-s + 0.852·22-s − 0.612·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s + 0.365·30-s + 0.883·32-s + 0.696·33-s + 1.02·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53067\)    =    \(3 \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(423.742\)
Root analytic conductor: \(20.5849\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 53067,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.006933317\)
\(L(\frac12)\) \(\approx\) \(5.006933317\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.26761730423118, −14.02654627006917, −13.64010916999055, −12.97184075444396, −12.56304279542886, −11.97870782124601, −11.74372722333197, −10.80386466804585, −10.00639723860365, −9.811077276806005, −9.354370502235291, −8.786916603904875, −8.302026918223375, −7.617726707216812, −6.970038957554238, −6.273341199405173, −5.945671186419837, −5.075089637508394, −4.944372456073412, −3.882913127442827, −3.652627363541610, −2.957947913454104, −2.206797924158938, −1.493602062690372, −0.6928302752923931, 0.6928302752923931, 1.493602062690372, 2.206797924158938, 2.957947913454104, 3.652627363541610, 3.882913127442827, 4.944372456073412, 5.075089637508394, 5.945671186419837, 6.273341199405173, 6.970038957554238, 7.617726707216812, 8.302026918223375, 8.786916603904875, 9.354370502235291, 9.811077276806005, 10.00639723860365, 10.80386466804585, 11.74372722333197, 11.97870782124601, 12.56304279542886, 12.97184075444396, 13.64010916999055, 14.02654627006917, 14.26761730423118

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