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Properties

Label 2-2040-2040.1109-c0-0-3
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  + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)6-s − i·8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + (−0.707 + 0.707i)12-s + 1.41·13-s + 1.00i·15-s + 16-s − i·17-s + 1.00·18-s + (0.707 − 0.707i)20-s + ⋯
L(s)  = 1  + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)6-s − i·8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + (−0.707 + 0.707i)12-s + 1.41·13-s + 1.00i·15-s + 16-s − i·17-s + 1.00·18-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: $\chi_{2040} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2040,\ (\ :0),\ 0.788 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248468891\)
\(L(\frac12)\) \(\approx\) \(1.248468891\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + iT \)
good7 \( 1 - iT^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - 1.41T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + (-1 + i)T - iT^{2} \)
37 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 - 2T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (1 + i)T + iT^{2} \)
83 \( 1 + T^{2} \)
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

−9.133397702211611600355797478456, −8.331769540936104923204875427142, −7.84681764172367147729507190556, −7.05380884070122702444287709903, −6.56857189516605254308522015996, −5.72892494726628488942530674530, −4.49680513097439473314905754589, −3.56791751927935149203697644466, −2.89965848028988088040174761889, −1.10731132304985474754128575653, 1.21884650021462173129695201942, 2.47331469007647382926837469520, 3.70876248132783035977367448230, 3.89641536094551759624369467164, 4.91244654380516258994763354782, 5.66762581524266877450205463682, 7.13590116390657978226553630620, 8.258620167047191786807995443237, 8.780944325209169975958455086589, 8.900384388675439326490535721591

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