Properties

Label 2-5054-1.1-c1-0-0
Degree $2$
Conductor $5054$
Sign $1$
Analytic cond. $40.3563$
Root an. cond. $6.35266$
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 + 0.00478·3-s + 4-s − 1.04·5-s − 0.00478·6-s − 7-s − 8-s − 2.99·9-s + 1.04·10-s − 4.80·11-s + 0.00478·12-s − 3.58·13-s + 14-s − 0.00497·15-s + 16-s − 6.75·17-s + 2.99·18-s − 1.04·20-s − 0.00478·21-s + 4.80·22-s + 0.904·23-s − 0.00478·24-s − 3.91·25-s + 3.58·26-s − 0.0287·27-s − 28-s − 3.14·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.00276·3-s + 0.5·4-s − 0.465·5-s − 0.00195·6-s − 0.377·7-s − 0.353·8-s − 0.999·9-s + 0.328·10-s − 1.44·11-s + 0.00138·12-s − 0.994·13-s + 0.267·14-s − 0.00128·15-s + 0.250·16-s − 1.63·17-s + 0.707·18-s − 0.232·20-s − 0.00104·21-s + 1.02·22-s + 0.188·23-s − 0.000976·24-s − 0.783·25-s + 0.703·26-s − 0.00552·27-s − 0.188·28-s − 0.584·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5054\)    =    \(2 \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(40.3563\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5054} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05855206362\)
\(L(\frac12)\) \(\approx\) \(0.05855206362\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
19 \( 1 \)
good3 \( 1 - 0.00478T + 3T^{2} \)
5 \( 1 + 1.04T + 5T^{2} \)
11 \( 1 + 4.80T + 11T^{2} \)
13 \( 1 + 3.58T + 13T^{2} \)
17 \( 1 + 6.75T + 17T^{2} \)
23 \( 1 - 0.904T + 23T^{2} \)
29 \( 1 + 3.14T + 29T^{2} \)
31 \( 1 + 2.02T + 31T^{2} \)
37 \( 1 + 9.28T + 37T^{2} \)
41 \( 1 + 8.12T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 + 8.76T + 53T^{2} \)
59 \( 1 - 1.50T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + 8.96T + 67T^{2} \)
71 \( 1 + 8.59T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 0.247T + 83T^{2} \)
89 \( 1 + 3.50T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−8.227913181777495054524184307879, −7.57899487932291126990397557198, −7.05620486258240764011408032640, −6.12774828559190400444138511809, −5.39347094706267338226837616377, −4.64093956525895803894817530468, −3.51451153377914709264976836960, −2.64999263848230237145121001354, −2.06569571516420468216385612774, −0.13304057559465360881475282462, 0.13304057559465360881475282462, 2.06569571516420468216385612774, 2.64999263848230237145121001354, 3.51451153377914709264976836960, 4.64093956525895803894817530468, 5.39347094706267338226837616377, 6.12774828559190400444138511809, 7.05620486258240764011408032640, 7.57899487932291126990397557198, 8.227913181777495054524184307879

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