Properties

Label 2-5054-1.1-c1-0-119
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 + 2.59·3-s + 4-s + 3.85·5-s − 2.59·6-s + 7-s − 8-s + 3.70·9-s − 3.85·10-s + 5.26·11-s + 2.59·12-s + 2.42·13-s − 14-s + 9.98·15-s + 16-s − 3.44·17-s − 3.70·18-s + 3.85·20-s + 2.59·21-s − 5.26·22-s − 5.54·23-s − 2.59·24-s + 9.86·25-s − 2.42·26-s + 1.83·27-s + 28-s − 9.13·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.49·3-s + 0.5·4-s + 1.72·5-s − 1.05·6-s + 0.377·7-s − 0.353·8-s + 1.23·9-s − 1.21·10-s + 1.58·11-s + 0.747·12-s + 0.673·13-s − 0.267·14-s + 2.57·15-s + 0.250·16-s − 0.834·17-s − 0.874·18-s + 0.861·20-s + 0.565·21-s − 1.12·22-s − 1.15·23-s − 0.528·24-s + 1.97·25-s − 0.476·26-s + 0.353·27-s + 0.188·28-s − 1.69·29-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

\(L(1)\) \(\approx\) \(4.060198059\)
\(L(\frac12)\) \(\approx\) \(4.060198059\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.59T + 3T^{2} \)
5 \( 1 - 3.85T + 5T^{2} \)
11 \( 1 - 5.26T + 11T^{2} \)
13 \( 1 - 2.42T + 13T^{2} \)
17 \( 1 + 3.44T + 17T^{2} \)
23 \( 1 + 5.54T + 23T^{2} \)
29 \( 1 + 9.13T + 29T^{2} \)
31 \( 1 + 6.65T + 31T^{2} \)
37 \( 1 - 0.133T + 37T^{2} \)
41 \( 1 - 6.10T + 41T^{2} \)
43 \( 1 - 9.10T + 43T^{2} \)
47 \( 1 - 2.28T + 47T^{2} \)
53 \( 1 - 5.31T + 53T^{2} \)
59 \( 1 - 6.81T + 59T^{2} \)
61 \( 1 + 1.44T + 61T^{2} \)
67 \( 1 + 5.82T + 67T^{2} \)
71 \( 1 + 8.66T + 71T^{2} \)
73 \( 1 - 9.83T + 73T^{2} \)
79 \( 1 + 0.632T + 79T^{2} \)
83 \( 1 + 5.21T + 83T^{2} \)
89 \( 1 - 2.58T + 89T^{2} \)
97 \( 1 + 1.58T + 97T^{2} \)
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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

−8.690261200692748530111565049763, −7.59772800409604809927116940567, −6.99884474384701548919163165888, −6.08371510502231905919670565882, −5.68919604565261845733702902710, −4.21393603350359542965469009276, −3.61941129485597040489987455028, −2.41430379354158142486526244007, −1.95264121133363215172493764349, −1.30601609936167490332960647576, 1.30601609936167490332960647576, 1.95264121133363215172493764349, 2.41430379354158142486526244007, 3.61941129485597040489987455028, 4.21393603350359542965469009276, 5.68919604565261845733702902710, 6.08371510502231905919670565882, 6.99884474384701548919163165888, 7.59772800409604809927116940567, 8.690261200692748530111565049763

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