Properties

Label 2-5054-1.1-c1-0-117
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 − 2.56·3-s + 4-s + 1.56·5-s − 2.56·6-s − 7-s + 8-s + 3.56·9-s + 1.56·10-s − 6.56·11-s − 2.56·12-s + 0.438·13-s − 14-s − 4·15-s + 16-s + 2·17-s + 3.56·18-s + 1.56·20-s + 2.56·21-s − 6.56·22-s + 3.56·23-s − 2.56·24-s − 2.56·25-s + 0.438·26-s − 1.43·27-s − 28-s + 10.2·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.47·3-s + 0.5·4-s + 0.698·5-s − 1.04·6-s − 0.377·7-s + 0.353·8-s + 1.18·9-s + 0.493·10-s − 1.97·11-s − 0.739·12-s + 0.121·13-s − 0.267·14-s − 1.03·15-s + 0.250·16-s + 0.485·17-s + 0.839·18-s + 0.349·20-s + 0.558·21-s − 1.39·22-s + 0.742·23-s − 0.522·24-s − 0.512·25-s + 0.0859·26-s − 0.276·27-s − 0.188·28-s + 1.90·29-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 \( 1 + 2.56T + 3T^{2} \)
5 \( 1 - 1.56T + 5T^{2} \)
11 \( 1 + 6.56T + 11T^{2} \)
13 \( 1 - 0.438T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
23 \( 1 - 3.56T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 - 7.68T + 41T^{2} \)
43 \( 1 + 7.12T + 43T^{2} \)
47 \( 1 + 7.12T + 47T^{2} \)
53 \( 1 + 9.12T + 53T^{2} \)
59 \( 1 + 11T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 - 3.68T + 67T^{2} \)
71 \( 1 + 5.80T + 71T^{2} \)
73 \( 1 - 4.56T + 73T^{2} \)
79 \( 1 - 2.24T + 79T^{2} \)
83 \( 1 + T + 83T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + 3.43T + 97T^{2} \)
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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

−7.67612270193975430926223395278, −6.83648146032189259053632856543, −6.08746778726754815952129415524, −5.79809960930207723972390539998, −4.86645487900528445764371927722, −4.73939206786607909020429387980, −3.20169500745029864967524845650, −2.56603192282416082164874650039, −1.28200103587936256617730006073, 0, 1.28200103587936256617730006073, 2.56603192282416082164874650039, 3.20169500745029864967524845650, 4.73939206786607909020429387980, 4.86645487900528445764371927722, 5.79809960930207723972390539998, 6.08746778726754815952129415524, 6.83648146032189259053632856543, 7.67612270193975430926223395278

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