Properties

Label 2-431-431.430-c0-0-1
Degree $2$
Conductor $431$
Sign $1$
Analytic cond. $0.215097$
Root an. cond. $0.463785$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.80·2-s − 0.445·3-s + 2.24·4-s − 1.80·5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s + 3.24·10-s + 2·11-s − 12-s + 0.801·15-s + 1.80·16-s + 1.44·18-s − 0.445·19-s − 4.04·20-s − 3.60·22-s + 1.24·23-s + 1.00·24-s + 2.24·25-s + 0.801·27-s + 1.24·29-s − 1.44·30-s − 1.00·32-s − 0.890·33-s − 1.80·36-s + 0.801·38-s + 4.04·40-s + ⋯
L(s)  = 1  − 1.80·2-s − 0.445·3-s + 2.24·4-s − 1.80·5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s + 3.24·10-s + 2·11-s − 12-s + 0.801·15-s + 1.80·16-s + 1.44·18-s − 0.445·19-s − 4.04·20-s − 3.60·22-s + 1.24·23-s + 1.00·24-s + 2.24·25-s + 0.801·27-s + 1.24·29-s − 1.44·30-s − 1.00·32-s − 0.890·33-s − 1.80·36-s + 0.801·38-s + 4.04·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(431\)
Sign: $1$
Analytic conductor: \(0.215097\)
Root analytic conductor: \(0.463785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (430, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 431,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2486380949\)
\(L(\frac12)\) \(\approx\) \(0.2486380949\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 - T \)
good2 \( 1 + 1.80T + T^{2} \)
3 \( 1 + 0.445T + T^{2} \)
5 \( 1 + 1.80T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 2T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 0.445T + T^{2} \)
23 \( 1 - 1.24T + T^{2} \)
29 \( 1 - 1.24T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.24T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 + 1.80T + T^{2} \)
61 \( 1 + 0.445T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−11.19228054737707333273482427874, −10.68712987451900361098159070559, −9.200620306338118211942704328064, −8.762861074425388030506493418900, −7.916432586930642498200672366598, −6.99697122923954466430808933502, −6.31790254194168164270036254696, −4.39876310194177161529601183170, −3.08646167109651913768329269533, −0.948072856839569317356302017586, 0.948072856839569317356302017586, 3.08646167109651913768329269533, 4.39876310194177161529601183170, 6.31790254194168164270036254696, 6.99697122923954466430808933502, 7.916432586930642498200672366598, 8.762861074425388030506493418900, 9.200620306338118211942704328064, 10.68712987451900361098159070559, 11.19228054737707333273482427874

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