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Properties

Label 2-2070-15.8-c1-0-24
Degree $2$
Conductor $2070$
Sign $0.608 - 0.793i$
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 + (1.46 − 1.68i)5-s + (−0.707 + 0.707i)8-s + (2.23 − 0.154i)10-s + 1.23i·11-s + (4.46 + 4.46i)13-s − 1.00·16-s + (4.78 + 4.78i)17-s − 5.84i·19-s + (1.68 + 1.46i)20-s + (−0.876 + 0.876i)22-s + (−0.707 + 0.707i)23-s + (−0.690 − 4.95i)25-s + 6.30i·26-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.656 − 0.754i)5-s + (−0.250 + 0.250i)8-s + (0.705 − 0.0489i)10-s + 0.373i·11-s + (1.23 + 1.23i)13-s − 0.250·16-s + (1.16 + 1.16i)17-s − 1.34i·19-s + (0.377 + 0.328i)20-s + (−0.186 + 0.186i)22-s + (−0.147 + 0.147i)23-s + (−0.138 − 0.990i)25-s + 1.23i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.608 - 0.793i$
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: \((2,\ 2070,\ (\ :1/2),\ 0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.810054215\)
\(L(\frac12)\) \(\approx\) \(2.810054215\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.46 + 1.68i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 - 1.23iT - 11T^{2} \)
13 \( 1 + (-4.46 - 4.46i)T + 13iT^{2} \)
17 \( 1 + (-4.78 - 4.78i)T + 17iT^{2} \)
19 \( 1 + 5.84iT - 19T^{2} \)
29 \( 1 + 4.52T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (-3.76 + 3.76i)T - 37iT^{2} \)
41 \( 1 + 8.85iT - 41T^{2} \)
43 \( 1 + (-7.64 - 7.64i)T + 43iT^{2} \)
47 \( 1 + (-1.69 - 1.69i)T + 47iT^{2} \)
53 \( 1 + (3.04 - 3.04i)T - 53iT^{2} \)
59 \( 1 - 0.693T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 + (-2.96 + 2.96i)T - 67iT^{2} \)
71 \( 1 - 16.2iT - 71T^{2} \)
73 \( 1 + (-4.66 - 4.66i)T + 73iT^{2} \)
79 \( 1 - 8.15iT - 79T^{2} \)
83 \( 1 + (-0.939 + 0.939i)T - 83iT^{2} \)
89 \( 1 - 4.52T + 89T^{2} \)
97 \( 1 + (-9.57 + 9.57i)T - 97iT^{2} \)
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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

−9.135313725604727156200506583282, −8.521951093703990611916719453892, −7.60570178652866140161946718501, −6.76174935437973460734133989743, −5.90732487406126382001312003438, −5.44367630433348808761352408155, −4.31263953694999185341140384499, −3.79451341568075988346277871614, −2.32231756566345054174154845922, −1.27235833311238239284524974951, 0.990183900534293718721412031807, 2.18655754003454611841440833392, 3.34428424526230127158946361440, 3.61559709657549812065108113630, 5.23405727360089933106431954264, 5.71771336743960231095768147733, 6.36381124399499974266207455540, 7.46691833610836111162258344131, 8.155850757830486768523732995361, 9.255249108281134479453943934363

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