Properties

Label 2-71058-1.1-c1-0-4
Degree $2$
Conductor $71058$
Sign $1$
Analytic cond. $567.400$
Root an. cond. $23.8201$
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 + 3·5-s + 6-s + 3·7-s − 8-s + 9-s − 3·10-s + 6·11-s − 12-s − 13-s − 3·14-s − 3·15-s + 16-s + 3·17-s − 18-s − 6·19-s + 3·20-s − 3·21-s − 6·22-s + 4·23-s + 24-s + 4·25-s + 26-s − 27-s + 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.80·11-s − 0.288·12-s − 0.277·13-s − 0.801·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.37·19-s + 0.670·20-s − 0.654·21-s − 1.27·22-s + 0.834·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s − 0.192·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71058\)    =    \(2 \cdot 3 \cdot 13 \cdot 911\)
Sign: $1$
Analytic conductor: \(567.400\)
Root analytic conductor: \(23.8201\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 71058,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.377039658\)
\(L(\frac12)\) \(\approx\) \(3.377039658\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + T \)
911 \( 1 - T \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−14.27154279075300, −13.75408815010075, −13.06195903908557, −12.48620720703736, −11.99820152631183, −11.55848570649913, −11.05588308953384, −10.55665326345015, −10.06650789582488, −9.569371548404967, −9.134009931321519, −8.568339928203348, −8.130203308518081, −7.369759779221020, −6.765738042432906, −6.275770116800401, −6.032790673824595, −5.223370186867399, −4.693303574550631, −4.170581670558521, −3.271493986115647, −2.335047192556437, −1.894654091430348, −1.198268171887339, −0.8229369829420488, 0.8229369829420488, 1.198268171887339, 1.894654091430348, 2.335047192556437, 3.271493986115647, 4.170581670558521, 4.693303574550631, 5.223370186867399, 6.032790673824595, 6.275770116800401, 6.765738042432906, 7.369759779221020, 8.130203308518081, 8.568339928203348, 9.134009931321519, 9.569371548404967, 10.06650789582488, 10.55665326345015, 11.05588308953384, 11.55848570649913, 11.99820152631183, 12.48620720703736, 13.06195903908557, 13.75408815010075, 14.27154279075300

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