Properties

Label 2-5070-1.1-c1-0-10
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 − 3.04·7-s − 8-s + 9-s + 10-s + 6.24·11-s − 12-s + 3.04·14-s + 15-s + 16-s + 2.69·17-s − 18-s + 5.82·19-s − 20-s + 3.04·21-s − 6.24·22-s − 5.62·23-s + 24-s + 25-s − 27-s − 3.04·28-s − 5.14·29-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 − 1.15·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.88·11-s − 0.288·12-s + 0.814·14-s + 0.258·15-s + 0.250·16-s + 0.652·17-s − 0.235·18-s + 1.33·19-s − 0.223·20-s + 0.665·21-s − 1.33·22-s − 1.17·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.576·28-s − 0.955·29-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

\(L(1)\) \(\approx\) \(0.9406230036\)
\(L(\frac12)\) \(\approx\) \(0.9406230036\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.04T + 7T^{2} \)
11 \( 1 - 6.24T + 11T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 - 5.82T + 19T^{2} \)
23 \( 1 + 5.62T + 23T^{2} \)
29 \( 1 + 5.14T + 29T^{2} \)
31 \( 1 - 3.53T + 31T^{2} \)
37 \( 1 - 4.65T + 37T^{2} \)
41 \( 1 - 3.77T + 41T^{2} \)
43 \( 1 - 2.85T + 43T^{2} \)
47 \( 1 - 3.61T + 47T^{2} \)
53 \( 1 - 0.664T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + 7.91T + 61T^{2} \)
67 \( 1 - 0.198T + 67T^{2} \)
71 \( 1 - 0.374T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + 6.83T + 83T^{2} \)
89 \( 1 + 9.50T + 89T^{2} \)
97 \( 1 - 0.335T + 97T^{2} \)
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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

−8.254310396839938562179579564069, −7.31075237809650076807888570509, −6.98665011183212563956694703331, −6.01678383167122941514830628513, −5.78257793118141653121458651401, −4.32964040596805463641252783431, −3.72314610947989989444636360494, −2.91737397049297718024868799860, −1.51440730829124522439258797641, −0.64740718895793163087470704205, 0.64740718895793163087470704205, 1.51440730829124522439258797641, 2.91737397049297718024868799860, 3.72314610947989989444636360494, 4.32964040596805463641252783431, 5.78257793118141653121458651401, 6.01678383167122941514830628513, 6.98665011183212563956694703331, 7.31075237809650076807888570509, 8.254310396839938562179579564069

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