Properties

Label 2-5054-1.1-c1-0-126
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 + 2.22·3-s + 4-s + 3.08·5-s + 2.22·6-s + 7-s + 8-s + 1.96·9-s + 3.08·10-s + 1.14·11-s + 2.22·12-s − 3.25·13-s + 14-s + 6.87·15-s + 16-s + 4.21·17-s + 1.96·18-s + 3.08·20-s + 2.22·21-s + 1.14·22-s − 0.445·23-s + 2.22·24-s + 4.52·25-s − 3.25·26-s − 2.30·27-s + 28-s − 9.88·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.28·3-s + 0.5·4-s + 1.38·5-s + 0.909·6-s + 0.377·7-s + 0.353·8-s + 0.654·9-s + 0.976·10-s + 0.345·11-s + 0.643·12-s − 0.903·13-s + 0.267·14-s + 1.77·15-s + 0.250·16-s + 1.02·17-s + 0.462·18-s + 0.690·20-s + 0.486·21-s + 0.244·22-s − 0.0929·23-s + 0.454·24-s + 0.905·25-s − 0.639·26-s − 0.444·27-s + 0.188·28-s − 1.83·29-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

\(L(1)\) \(\approx\) \(6.819937813\)
\(L(\frac12)\) \(\approx\) \(6.819937813\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.22T + 3T^{2} \)
5 \( 1 - 3.08T + 5T^{2} \)
11 \( 1 - 1.14T + 11T^{2} \)
13 \( 1 + 3.25T + 13T^{2} \)
17 \( 1 - 4.21T + 17T^{2} \)
23 \( 1 + 0.445T + 23T^{2} \)
29 \( 1 + 9.88T + 29T^{2} \)
31 \( 1 - 1.67T + 31T^{2} \)
37 \( 1 - 8.76T + 37T^{2} \)
41 \( 1 - 4.91T + 41T^{2} \)
43 \( 1 - 4.44T + 43T^{2} \)
47 \( 1 - 2.20T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 6.58T + 59T^{2} \)
61 \( 1 - 7.80T + 61T^{2} \)
67 \( 1 + 0.524T + 67T^{2} \)
71 \( 1 - 4.81T + 71T^{2} \)
73 \( 1 - 8.13T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−7.990307189842674929865753792861, −7.69530580402624176840095229109, −6.77680954866613224155642040855, −5.84353554205105654504987420601, −5.44106846211082356328644834417, −4.45399258682087081438724612355, −3.63134259965231073941765289650, −2.72333235765668763068639052072, −2.20882012673850199813727605203, −1.38888642801720435226278646263, 1.38888642801720435226278646263, 2.20882012673850199813727605203, 2.72333235765668763068639052072, 3.63134259965231073941765289650, 4.45399258682087081438724612355, 5.44106846211082356328644834417, 5.84353554205105654504987420601, 6.77680954866613224155642040855, 7.69530580402624176840095229109, 7.990307189842674929865753792861

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