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Properties

Label 2-2070-15.8-c1-0-25
Degree $2$
Conductor $2070$
Sign $-0.0918 + 0.995i$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.342 + 2.20i)5-s + (−2 + 2i)7-s + (0.707 − 0.707i)8-s + (1.80 − 1.32i)10-s − 1.59i·11-s + (−2.64 − 2.64i)13-s + 2.82·14-s − 1.00·16-s + (−1.41 − 1.41i)17-s + 1.60i·19-s + (−2.20 − 0.342i)20-s + (−1.12 + 1.12i)22-s + (0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.153 + 0.988i)5-s + (−0.755 + 0.755i)7-s + (0.250 − 0.250i)8-s + (0.570 − 0.417i)10-s − 0.479i·11-s + (−0.732 − 0.732i)13-s + 0.755·14-s − 0.250·16-s + (−0.342 − 0.342i)17-s + 0.369i·19-s + (−0.494 − 0.0766i)20-s + (−0.239 + 0.239i)22-s + (0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0918 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0918 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.0918 + 0.995i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ -0.0918 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5621578040\)
\(L(\frac12)\) \(\approx\) \(0.5621578040\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (0.342 - 2.20i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 + 1.59iT - 11T^{2} \)
13 \( 1 + (2.64 + 2.64i)T + 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 - 1.60iT - 19T^{2} \)
29 \( 1 - 6.56T + 29T^{2} \)
31 \( 1 + 4.24T + 31T^{2} \)
37 \( 1 + (0.875 - 0.875i)T - 37iT^{2} \)
41 \( 1 - 2.87iT - 41T^{2} \)
43 \( 1 + (2.09 + 2.09i)T + 43iT^{2} \)
47 \( 1 + (3.51 + 3.51i)T + 47iT^{2} \)
53 \( 1 + (-3.23 + 3.23i)T - 53iT^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 + 9.85T + 61T^{2} \)
67 \( 1 + (-4.79 + 4.79i)T - 67iT^{2} \)
71 \( 1 + 8.32iT - 71T^{2} \)
73 \( 1 + (2.03 + 2.03i)T + 73iT^{2} \)
79 \( 1 + 7.21iT - 79T^{2} \)
83 \( 1 + (-8.60 + 8.60i)T - 83iT^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + (-1.60 + 1.60i)T - 97iT^{2} \)
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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

−9.002689244890422959823931851662, −8.227266363773313613792579198489, −7.42719512059434378687961028992, −6.62947179680011903137287059477, −5.91782330511265610560317962937, −4.84991444913130753921449637666, −3.49419672497258620566523322628, −2.96595046792315170200715101395, −2.13047790876019524621485477127, −0.27823752113009139473362365561, 0.996225636689926100504869221834, 2.26243898008507419578267308626, 3.77394124961365344833718593221, 4.56913277164214633704131614113, 5.29691253135469223840179509808, 6.43580307276010162751136824055, 7.02022046380733815187241508047, 7.73682204816785540723459251390, 8.626806131737553456797371350413, 9.274979499916273683965429535696

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