Properties

Label 2-2070-1.1-c1-0-3
Degree $2$
Conductor $2070$
Sign $1$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
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 + 4-s − 5-s − 0.618·7-s − 8-s + 10-s + 2.85·11-s − 7.09·13-s + 0.618·14-s + 16-s − 6.09·17-s + 1.85·19-s − 20-s − 2.85·22-s − 23-s + 25-s + 7.09·26-s − 0.618·28-s + 9.23·29-s + 9.09·31-s − 32-s + 6.09·34-s + 0.618·35-s + 6.47·37-s − 1.85·38-s + 40-s − 3.32·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.233·7-s − 0.353·8-s + 0.316·10-s + 0.860·11-s − 1.96·13-s + 0.165·14-s + 0.250·16-s − 1.47·17-s + 0.425·19-s − 0.223·20-s − 0.608·22-s − 0.208·23-s + 0.200·25-s + 1.39·26-s − 0.116·28-s + 1.71·29-s + 1.63·31-s − 0.176·32-s + 1.04·34-s + 0.104·35-s + 1.06·37-s − 0.300·38-s + 0.158·40-s − 0.519·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8933470998\)
\(L(\frac12)\) \(\approx\) \(0.8933470998\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
23 \( 1 + T \)
good7 \( 1 + 0.618T + 7T^{2} \)
11 \( 1 - 2.85T + 11T^{2} \)
13 \( 1 + 7.09T + 13T^{2} \)
17 \( 1 + 6.09T + 17T^{2} \)
19 \( 1 - 1.85T + 19T^{2} \)
29 \( 1 - 9.23T + 29T^{2} \)
31 \( 1 - 9.09T + 31T^{2} \)
37 \( 1 - 6.47T + 37T^{2} \)
41 \( 1 + 3.32T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 3.70T + 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 + 1.70T + 59T^{2} \)
61 \( 1 + 9.32T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 - 4.09T + 71T^{2} \)
73 \( 1 - 3.23T + 73T^{2} \)
79 \( 1 - 1.52T + 79T^{2} \)
83 \( 1 - 6.94T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−9.223470128264673063072497875000, −8.326617505517797241380144977711, −7.67707462950570956460978681126, −6.74188919941959306768373955162, −6.41471522684785427656491994896, −4.92810056619173249586244672859, −4.34509925821010978980810211039, −3.01374878089399841621000142033, −2.20303926815927969478566357017, −0.67445417495635941704012673035, 0.67445417495635941704012673035, 2.20303926815927969478566357017, 3.01374878089399841621000142033, 4.34509925821010978980810211039, 4.92810056619173249586244672859, 6.41471522684785427656491994896, 6.74188919941959306768373955162, 7.67707462950570956460978681126, 8.326617505517797241380144977711, 9.223470128264673063072497875000

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