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Properties

Label 2-2040-2040.149-c0-0-4
Degree $2$
Conductor $2040$
Sign $0.788 + 0.615i$
Analytic cond. $1.01809$
Root an. cond. $1.00900$
Motivic weight $0$
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 + (−0.707 + 0.707i)3-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + 1.00i·10-s + (0.707 + 0.707i)12-s − 1.00i·15-s − 1.00·16-s + (0.707 − 0.707i)17-s + (−0.707 − 0.707i)18-s + 2·19-s + (0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + 1.00i·10-s + (0.707 + 0.707i)12-s − 1.00i·15-s − 1.00·16-s + (0.707 − 0.707i)17-s + (−0.707 − 0.707i)18-s + 2·19-s + (0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(1.01809\)
Root analytic conductor: \(1.00900\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2040} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2040,\ (\ :0),\ 0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.192924992\)
\(L(\frac12)\) \(\approx\) \(1.192924992\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - 2T + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (-1 - i)T + iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (1 - i)T - iT^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.718258863889934904931251155970, −8.692022249370655907763812741639, −7.39302768188221662452498232728, −6.79447248399121979990047683445, −5.74591642980833778795970160965, −5.15727533963024799891754739554, −4.28025893462848953192857959553, −3.40816352614912614945183492434, −2.84794460732860150303929464391, −0.971369189091242473228766705250, 1.14680795284069934943350413118, 2.78680175636179706835311034909, 3.90570255325368090170737682957, 4.73540172957647924361311337847, 5.57042967121399380550826617633, 6.01940747261519082479683284439, 7.25546641733798044346397498509, 7.56109370740969099427396206573, 8.268160146940717841540295317317, 9.124645424878434079975137279439

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