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Properties

Label 2-2070-15.8-c1-0-0
Degree $2$
Conductor $2070$
Sign $-0.998 - 0.0532i$
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 + (−2.22 − 0.256i)5-s + (−2 + 2i)7-s + (0.707 − 0.707i)8-s + (1.38 + 1.75i)10-s + 3.34i·11-s + (3.50 + 3.50i)13-s + 2.82·14-s − 1.00·16-s + (−1.41 − 1.41i)17-s + 0.778i·19-s + (0.256 − 2.22i)20-s + (2.36 − 2.36i)22-s + (0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.993 − 0.114i)5-s + (−0.755 + 0.755i)7-s + (0.250 − 0.250i)8-s + (0.439 + 0.554i)10-s + 1.00i·11-s + (0.972 + 0.972i)13-s + 0.755·14-s − 0.250·16-s + (−0.342 − 0.342i)17-s + 0.178i·19-s + (0.0574 − 0.496i)20-s + (0.503 − 0.503i)22-s + (0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.998 - 0.0532i$
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: \((2,\ 2070,\ (\ :1/2),\ -0.998 - 0.0532i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1426346401\)
\(L(\frac12)\) \(\approx\) \(0.1426346401\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.256i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 - 3.34iT - 11T^{2} \)
13 \( 1 + (-3.50 - 3.50i)T + 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 - 0.778iT - 19T^{2} \)
29 \( 1 + 2.12T + 29T^{2} \)
31 \( 1 - 2.72T + 31T^{2} \)
37 \( 1 + (4.36 - 4.36i)T - 37iT^{2} \)
41 \( 1 + 4.64iT - 41T^{2} \)
43 \( 1 + (3.91 + 3.91i)T + 43iT^{2} \)
47 \( 1 + (7.27 + 7.27i)T + 47iT^{2} \)
53 \( 1 + (-6.40 + 6.40i)T - 53iT^{2} \)
59 \( 1 - 1.80T + 59T^{2} \)
61 \( 1 + 2.05T + 61T^{2} \)
67 \( 1 + (10.1 - 10.1i)T - 67iT^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + (-3.28 - 3.28i)T + 73iT^{2} \)
79 \( 1 + 5.55iT - 79T^{2} \)
83 \( 1 + (9.36 - 9.36i)T - 83iT^{2} \)
89 \( 1 + 9.21T + 89T^{2} \)
97 \( 1 + (-0.778 + 0.778i)T - 97iT^{2} \)
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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

−9.455397181408538281534126482982, −8.695979242899071484818910376328, −8.287956551093321928472501704687, −7.00364044981605688277370805471, −6.79569321591152584985083157221, −5.46623674148934976307073850968, −4.38223663511665598146469038473, −3.69318593831866341721863732423, −2.70777530656179265924317043130, −1.58832498791433798613293901691, 0.07093399383835831983712895047, 1.10270263062870407261602264468, 3.07587139726363027338704406722, 3.63587529321207921343211325821, 4.65885102180051049932594559920, 5.85104315523056331301239081651, 6.45052145210801582776070964001, 7.27561609839875570395853466510, 8.021832274860268899001803329462, 8.534487940866355439681421515099

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