Properties

Label 2-5054-1.1-c1-0-114
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.34·3-s + 4-s − 2.87·5-s − 1.34·6-s + 7-s − 8-s − 1.18·9-s + 2.87·10-s − 0.652·11-s + 1.34·12-s + 1.53·13-s − 14-s − 3.87·15-s + 16-s + 5.63·17-s + 1.18·18-s − 2.87·20-s + 1.34·21-s + 0.652·22-s + 0.369·23-s − 1.34·24-s + 3.29·25-s − 1.53·26-s − 5.63·27-s + 28-s + 3.77·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.777·3-s + 0.5·4-s − 1.28·5-s − 0.550·6-s + 0.377·7-s − 0.353·8-s − 0.394·9-s + 0.910·10-s − 0.196·11-s + 0.388·12-s + 0.424·13-s − 0.267·14-s − 1.00·15-s + 0.250·16-s + 1.36·17-s + 0.279·18-s − 0.643·20-s + 0.294·21-s + 0.139·22-s + 0.0770·23-s − 0.275·24-s + 0.658·25-s − 0.300·26-s − 1.08·27-s + 0.188·28-s + 0.700·29-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: \(1\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.34T + 3T^{2} \)
5 \( 1 + 2.87T + 5T^{2} \)
11 \( 1 + 0.652T + 11T^{2} \)
13 \( 1 - 1.53T + 13T^{2} \)
17 \( 1 - 5.63T + 17T^{2} \)
23 \( 1 - 0.369T + 23T^{2} \)
29 \( 1 - 3.77T + 29T^{2} \)
31 \( 1 + 2.04T + 31T^{2} \)
37 \( 1 + 7.80T + 37T^{2} \)
41 \( 1 + 12.7T + 41T^{2} \)
43 \( 1 + 9.45T + 43T^{2} \)
47 \( 1 - 4.18T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 8.39T + 59T^{2} \)
61 \( 1 + 2.24T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 5.66T + 79T^{2} \)
83 \( 1 - 0.403T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 3.51T + 97T^{2} \)
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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

−8.211826329426982325368248771049, −7.38929306430245737405458067310, −6.83139639748571341907716405261, −5.67212422220794682046227082179, −4.96140584927743854577816492776, −3.63440956760380372215173011894, −3.47484667335726850016520009413, −2.39334166901707793521730267709, −1.27112120808680496365448716562, 0, 1.27112120808680496365448716562, 2.39334166901707793521730267709, 3.47484667335726850016520009413, 3.63440956760380372215173011894, 4.96140584927743854577816492776, 5.67212422220794682046227082179, 6.83139639748571341907716405261, 7.38929306430245737405458067310, 8.211826329426982325368248771049

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