Properties

Label 2-5070-1.1-c1-0-3
Degree $2$
Conductor $5070$
Sign $1$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s − 5·11-s − 12-s + 2·14-s − 15-s + 16-s − 2·17-s − 18-s − 2·19-s + 20-s + 2·21-s + 5·22-s − 23-s + 24-s + 25-s − 27-s − 2·28-s + 5·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s − 0.288·12-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.436·21-s + 1.06·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 0.928·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5300685247\)
\(L(\frac12)\) \(\approx\) \(0.5300685247\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.296426376373495940607688736053, −7.45347874572048090543347211913, −6.84006745824056260728561119029, −6.12931816980714195615570039074, −5.48507792110898740496492362388, −4.74849023536736863756449317184, −3.57579855417412730736535244824, −2.63787482548952120485166197373, −1.85595605838881399665339828644, −0.43210380631311830924336142990, 0.43210380631311830924336142990, 1.85595605838881399665339828644, 2.63787482548952120485166197373, 3.57579855417412730736535244824, 4.74849023536736863756449317184, 5.48507792110898740496492362388, 6.12931816980714195615570039074, 6.84006745824056260728561119029, 7.45347874572048090543347211913, 8.296426376373495940607688736053

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