Properties

Label 2-5054-1.1-c1-0-141
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 + 3.30·3-s + 4-s + 1.60·5-s + 3.30·6-s − 7-s + 8-s + 7.92·9-s + 1.60·10-s − 1.00·11-s + 3.30·12-s + 3.99·13-s − 14-s + 5.30·15-s + 16-s + 4.17·17-s + 7.92·18-s + 1.60·20-s − 3.30·21-s − 1.00·22-s − 3.41·23-s + 3.30·24-s − 2.42·25-s + 3.99·26-s + 16.2·27-s − 28-s − 9.28·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.90·3-s + 0.5·4-s + 0.718·5-s + 1.34·6-s − 0.377·7-s + 0.353·8-s + 2.64·9-s + 0.507·10-s − 0.303·11-s + 0.954·12-s + 1.10·13-s − 0.267·14-s + 1.37·15-s + 0.250·16-s + 1.01·17-s + 1.86·18-s + 0.359·20-s − 0.721·21-s − 0.214·22-s − 0.711·23-s + 0.674·24-s − 0.484·25-s + 0.783·26-s + 3.13·27-s − 0.188·28-s − 1.72·29-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

\(L(1)\) \(\approx\) \(7.561169024\)
\(L(\frac12)\) \(\approx\) \(7.561169024\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.30T + 3T^{2} \)
5 \( 1 - 1.60T + 5T^{2} \)
11 \( 1 + 1.00T + 11T^{2} \)
13 \( 1 - 3.99T + 13T^{2} \)
17 \( 1 - 4.17T + 17T^{2} \)
23 \( 1 + 3.41T + 23T^{2} \)
29 \( 1 + 9.28T + 29T^{2} \)
31 \( 1 - 3.52T + 31T^{2} \)
37 \( 1 - 6.29T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 + 3.45T + 43T^{2} \)
47 \( 1 + 6.77T + 47T^{2} \)
53 \( 1 + 0.844T + 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 + 2.17T + 61T^{2} \)
67 \( 1 - 1.83T + 67T^{2} \)
71 \( 1 - 1.00T + 71T^{2} \)
73 \( 1 - 4.42T + 73T^{2} \)
79 \( 1 - 3.58T + 79T^{2} \)
83 \( 1 - 9.90T + 83T^{2} \)
89 \( 1 - 0.861T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−7.988840782203004016375910317530, −7.84044152751627568278844797112, −6.76788488215794018538707287503, −6.12330256913384526812218484524, −5.27134265900891318835413628685, −4.22219787024026446377393407892, −3.50095399373377570906038160739, −3.08373349665684541916723743671, −2.05934266055957602405599695513, −1.49311838123544884429564688706, 1.49311838123544884429564688706, 2.05934266055957602405599695513, 3.08373349665684541916723743671, 3.50095399373377570906038160739, 4.22219787024026446377393407892, 5.27134265900891318835413628685, 6.12330256913384526812218484524, 6.76788488215794018538707287503, 7.84044152751627568278844797112, 7.988840782203004016375910317530

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