Properties

Label 2-5054-1.1-c1-0-129
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 + 2.52·3-s + 4-s + 1.42·5-s + 2.52·6-s + 7-s + 8-s + 3.36·9-s + 1.42·10-s + 1.22·11-s + 2.52·12-s + 2.87·13-s + 14-s + 3.58·15-s + 16-s + 3.92·17-s + 3.36·18-s + 1.42·20-s + 2.52·21-s + 1.22·22-s − 3.82·23-s + 2.52·24-s − 2.98·25-s + 2.87·26-s + 0.913·27-s + 28-s − 1.85·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.45·3-s + 0.5·4-s + 0.635·5-s + 1.02·6-s + 0.377·7-s + 0.353·8-s + 1.12·9-s + 0.449·10-s + 0.368·11-s + 0.728·12-s + 0.798·13-s + 0.267·14-s + 0.925·15-s + 0.250·16-s + 0.951·17-s + 0.792·18-s + 0.317·20-s + 0.550·21-s + 0.260·22-s − 0.796·23-s + 0.514·24-s − 0.596·25-s + 0.564·26-s + 0.175·27-s + 0.188·28-s − 0.344·29-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

\(L(1)\) \(\approx\) \(6.774905045\)
\(L(\frac12)\) \(\approx\) \(6.774905045\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.52T + 3T^{2} \)
5 \( 1 - 1.42T + 5T^{2} \)
11 \( 1 - 1.22T + 11T^{2} \)
13 \( 1 - 2.87T + 13T^{2} \)
17 \( 1 - 3.92T + 17T^{2} \)
23 \( 1 + 3.82T + 23T^{2} \)
29 \( 1 + 1.85T + 29T^{2} \)
31 \( 1 + 4.99T + 31T^{2} \)
37 \( 1 + 0.841T + 37T^{2} \)
41 \( 1 - 7.37T + 41T^{2} \)
43 \( 1 + 4.80T + 43T^{2} \)
47 \( 1 - 6.55T + 47T^{2} \)
53 \( 1 - 3.68T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 + 9.35T + 67T^{2} \)
71 \( 1 - 9.04T + 71T^{2} \)
73 \( 1 + 4.26T + 73T^{2} \)
79 \( 1 + 1.87T + 79T^{2} \)
83 \( 1 + 3.84T + 83T^{2} \)
89 \( 1 - 18.4T + 89T^{2} \)
97 \( 1 + 17.7T + 97T^{2} \)
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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

−8.072199684119871973653441795365, −7.71041384118449288546972758637, −6.83006664992395322140180015424, −5.88187895789188215410345673898, −5.44868008017565990980151219728, −4.17736834438346434348123148068, −3.75710988852325522403659470038, −2.91799296932183364607204287231, −2.06956016977621900552410398895, −1.40074981971049265504399691178, 1.40074981971049265504399691178, 2.06956016977621900552410398895, 2.91799296932183364607204287231, 3.75710988852325522403659470038, 4.17736834438346434348123148068, 5.44868008017565990980151219728, 5.88187895789188215410345673898, 6.83006664992395322140180015424, 7.71041384118449288546972758637, 8.072199684119871973653441795365

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