Properties

Label 2-71058-1.1-c1-0-9
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 2·7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 2·14-s − 15-s + 16-s + 3·17-s + 18-s + 7·19-s − 20-s − 2·21-s + 22-s − 4·23-s + 24-s − 4·25-s + 26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 1.60·19-s − 0.223·20-s − 0.436·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.377·28-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: \(1\)
Selberg data: \((2,\ 71058,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
13 \( 1 - T \)
911 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−14.32028625774871, −13.96952594437785, −13.35775355524712, −12.97337341467782, −12.39323645572314, −12.01965936205811, −11.46404664367509, −11.01245444754360, −10.31145428739793, −9.749951756896713, −9.307880753096565, −8.964432506687106, −7.913199602710229, −7.546837915179151, −7.469916473873707, −6.445426000129157, −6.005286266652099, −5.552423698125427, −4.820513932821583, −4.020596577635311, −3.722255133785634, −3.197622742896585, −2.636299954049761, −1.771994594212414, −1.102190149934339, 0, 1.102190149934339, 1.771994594212414, 2.636299954049761, 3.197622742896585, 3.722255133785634, 4.020596577635311, 4.820513932821583, 5.552423698125427, 6.005286266652099, 6.445426000129157, 7.469916473873707, 7.546837915179151, 7.913199602710229, 8.964432506687106, 9.307880753096565, 9.749951756896713, 10.31145428739793, 11.01245444754360, 11.46404664367509, 12.01965936205811, 12.39323645572314, 12.97337341467782, 13.35775355524712, 13.96952594437785, 14.32028625774871

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