Properties

Label 2-574-1.1-c1-0-0
Degree $2$
Conductor $574$
Sign $1$
Analytic cond. $4.58341$
Root an. cond. $2.14089$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.12·3-s + 4-s − 3.70·5-s + 1.12·6-s − 7-s − 8-s − 1.73·9-s + 3.70·10-s + 4.57·11-s − 1.12·12-s − 5.55·13-s + 14-s + 4.17·15-s + 16-s + 1.43·17-s + 1.73·18-s − 3.55·19-s − 3.70·20-s + 1.12·21-s − 4.57·22-s + 5.55·23-s + 1.12·24-s + 8.72·25-s + 5.55·26-s + 5.32·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.650·3-s + 0.5·4-s − 1.65·5-s + 0.459·6-s − 0.377·7-s − 0.353·8-s − 0.577·9-s + 1.17·10-s + 1.38·11-s − 0.325·12-s − 1.53·13-s + 0.267·14-s + 1.07·15-s + 0.250·16-s + 0.347·17-s + 0.408·18-s − 0.814·19-s − 0.828·20-s + 0.245·21-s − 0.976·22-s + 1.15·23-s + 0.229·24-s + 1.74·25-s + 1.08·26-s + 1.02·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(574\)    =    \(2 \cdot 7 \cdot 41\)
Sign: $1$
Analytic conductor: \(4.58341\)
Root analytic conductor: \(2.14089\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 574,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4278295318\)
\(L(\frac12)\) \(\approx\) \(0.4278295318\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
41 \( 1 - T \)
good3 \( 1 + 1.12T + 3T^{2} \)
5 \( 1 + 3.70T + 5T^{2} \)
11 \( 1 - 4.57T + 11T^{2} \)
13 \( 1 + 5.55T + 13T^{2} \)
17 \( 1 - 1.43T + 17T^{2} \)
19 \( 1 + 3.55T + 19T^{2} \)
23 \( 1 - 5.55T + 23T^{2} \)
29 \( 1 - 6.40T + 29T^{2} \)
31 \( 1 + 0.172T + 31T^{2} \)
37 \( 1 - 0.252T + 37T^{2} \)
43 \( 1 + 6.42T + 43T^{2} \)
47 \( 1 - 8.11T + 47T^{2} \)
53 \( 1 + 8.41T + 53T^{2} \)
59 \( 1 - 9.87T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 7.13T + 67T^{2} \)
71 \( 1 - 5.81T + 71T^{2} \)
73 \( 1 - 1.38T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 - 9.58T + 97T^{2} \)
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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

−10.88666376344184805062850950743, −9.870666663075032552776576265569, −8.856896110991864497084457984185, −8.160810837509841927419448417816, −7.06274955828220202435048597931, −6.60084233518333489883423935489, −5.12676978873747803931935904066, −4.04113978157380851187750642065, −2.85321761632502117640457185816, −0.63967045228762778715899693228, 0.63967045228762778715899693228, 2.85321761632502117640457185816, 4.04113978157380851187750642065, 5.12676978873747803931935904066, 6.60084233518333489883423935489, 7.06274955828220202435048597931, 8.160810837509841927419448417816, 8.856896110991864497084457984185, 9.870666663075032552776576265569, 10.88666376344184805062850950743

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