Properties

Label 2-5070-1.1-c1-0-14
Degree $2$
Conductor $5070$
Sign $1$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
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 + 3-s + 4-s − 5-s − 6-s + 2.82·7-s − 8-s + 9-s + 10-s − 5.65·11-s + 12-s − 2.82·14-s − 15-s + 16-s + 0.828·17-s − 18-s − 2.82·19-s − 20-s + 2.82·21-s + 5.65·22-s − 8.48·23-s − 24-s + 25-s + 27-s + 2.82·28-s − 8.82·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.06·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.70·11-s + 0.288·12-s − 0.755·14-s − 0.258·15-s + 0.250·16-s + 0.200·17-s − 0.235·18-s − 0.648·19-s − 0.223·20-s + 0.617·21-s + 1.20·22-s − 1.76·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.534·28-s − 1.63·29-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: no
Arithmetic: yes
Character: $\chi_{5070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.384574472\)
\(L(\frac12)\) \(\approx\) \(1.384574472\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
17 \( 1 - 0.828T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 8.48T + 23T^{2} \)
29 \( 1 + 8.82T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 - 7.65T + 41T^{2} \)
43 \( 1 - 9.65T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 - 2.34T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 14.4T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 + 6.34T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 - 3.17T + 97T^{2} \)
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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

−8.046518096152990333473559459369, −7.71850915730465861686981178830, −7.35197640108061847211878588373, −5.99352127903112958443237714312, −5.44264134928554155567223675013, −4.38658921926225356379316289599, −3.74989509798311547860657069625, −2.38710490289755682511670590389, −2.16089287176042850360167703122, −0.66933501506302940607173579969, 0.66933501506302940607173579969, 2.16089287176042850360167703122, 2.38710490289755682511670590389, 3.74989509798311547860657069625, 4.38658921926225356379316289599, 5.44264134928554155567223675013, 5.99352127903112958443237714312, 7.35197640108061847211878588373, 7.71850915730465861686981178830, 8.046518096152990333473559459369

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