Properties

Label 2-5054-1.1-c1-0-121
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 − 0.717·3-s + 4-s + 2.91·5-s + 0.717·6-s − 7-s − 8-s − 2.48·9-s − 2.91·10-s + 2.56·11-s − 0.717·12-s − 1.75·13-s + 14-s − 2.09·15-s + 16-s + 1.39·17-s + 2.48·18-s + 2.91·20-s + 0.717·21-s − 2.56·22-s − 3.44·23-s + 0.717·24-s + 3.50·25-s + 1.75·26-s + 3.93·27-s − 28-s + 0.789·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.414·3-s + 0.5·4-s + 1.30·5-s + 0.293·6-s − 0.377·7-s − 0.353·8-s − 0.828·9-s − 0.922·10-s + 0.773·11-s − 0.207·12-s − 0.487·13-s + 0.267·14-s − 0.540·15-s + 0.250·16-s + 0.338·17-s + 0.585·18-s + 0.652·20-s + 0.156·21-s − 0.546·22-s − 0.717·23-s + 0.146·24-s + 0.700·25-s + 0.344·26-s + 0.757·27-s − 0.188·28-s + 0.146·29-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 \( 1 + 0.717T + 3T^{2} \)
5 \( 1 - 2.91T + 5T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 + 1.75T + 13T^{2} \)
17 \( 1 - 1.39T + 17T^{2} \)
23 \( 1 + 3.44T + 23T^{2} \)
29 \( 1 - 0.789T + 29T^{2} \)
31 \( 1 + 4.99T + 31T^{2} \)
37 \( 1 + 7.90T + 37T^{2} \)
41 \( 1 + 6.60T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + 1.23T + 47T^{2} \)
53 \( 1 - 7.30T + 53T^{2} \)
59 \( 1 + 1.68T + 59T^{2} \)
61 \( 1 - 4.80T + 61T^{2} \)
67 \( 1 + 2.63T + 67T^{2} \)
71 \( 1 + 8.70T + 71T^{2} \)
73 \( 1 + 4.32T + 73T^{2} \)
79 \( 1 - 4.23T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 1.50T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−7.953472183870087397123582365788, −6.98971777392620322833019365994, −6.49729012742200083185006016561, −5.63287439539269870584735148868, −5.45599702649994615944450144268, −4.09547082285807320923848340297, −3.03100948467862119741942205117, −2.21365308701668306501813740112, −1.33576941047208968213458936862, 0, 1.33576941047208968213458936862, 2.21365308701668306501813740112, 3.03100948467862119741942205117, 4.09547082285807320923848340297, 5.45599702649994615944450144268, 5.63287439539269870584735148868, 6.49729012742200083185006016561, 6.98971777392620322833019365994, 7.953472183870087397123582365788

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