Properties

Label 2-5054-1.1-c1-0-122
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 + 2.30·3-s + 4-s + 3.15·5-s + 2.30·6-s − 7-s + 8-s + 2.31·9-s + 3.15·10-s − 2.86·11-s + 2.30·12-s + 6.74·13-s − 14-s + 7.26·15-s + 16-s − 3.61·17-s + 2.31·18-s + 3.15·20-s − 2.30·21-s − 2.86·22-s + 9.19·23-s + 2.30·24-s + 4.93·25-s + 6.74·26-s − 1.58·27-s − 28-s − 3.62·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.33·3-s + 0.5·4-s + 1.40·5-s + 0.941·6-s − 0.377·7-s + 0.353·8-s + 0.771·9-s + 0.996·10-s − 0.863·11-s + 0.665·12-s + 1.87·13-s − 0.267·14-s + 1.87·15-s + 0.250·16-s − 0.877·17-s + 0.545·18-s + 0.704·20-s − 0.503·21-s − 0.610·22-s + 1.91·23-s + 0.470·24-s + 0.986·25-s + 1.32·26-s − 0.304·27-s − 0.188·28-s − 0.673·29-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

\(L(1)\) \(\approx\) \(6.603134358\)
\(L(\frac12)\) \(\approx\) \(6.603134358\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.30T + 3T^{2} \)
5 \( 1 - 3.15T + 5T^{2} \)
11 \( 1 + 2.86T + 11T^{2} \)
13 \( 1 - 6.74T + 13T^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
23 \( 1 - 9.19T + 23T^{2} \)
29 \( 1 + 3.62T + 29T^{2} \)
31 \( 1 - 0.789T + 31T^{2} \)
37 \( 1 + 8.66T + 37T^{2} \)
41 \( 1 - 1.81T + 41T^{2} \)
43 \( 1 - 5.87T + 43T^{2} \)
47 \( 1 - 5.19T + 47T^{2} \)
53 \( 1 - 0.209T + 53T^{2} \)
59 \( 1 - 13.1T + 59T^{2} \)
61 \( 1 - 1.83T + 61T^{2} \)
67 \( 1 + 15.0T + 67T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 + 6.67T + 73T^{2} \)
79 \( 1 + 1.08T + 79T^{2} \)
83 \( 1 + 8.02T + 83T^{2} \)
89 \( 1 + 5.21T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−8.514366572565379911060564019925, −7.37597991961273798526271948386, −6.79389759196704213442188662768, −5.87247570302663585031157033847, −5.51387915035665797712233201913, −4.41490661429140413345965212211, −3.50209617169940595015736074326, −2.87413120421570406222440793668, −2.21003860126543491646618266564, −1.33851264462515067686884104439, 1.33851264462515067686884104439, 2.21003860126543491646618266564, 2.87413120421570406222440793668, 3.50209617169940595015736074326, 4.41490661429140413345965212211, 5.51387915035665797712233201913, 5.87247570302663585031157033847, 6.79389759196704213442188662768, 7.37597991961273798526271948386, 8.514366572565379911060564019925

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