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Properties

Label 2-2040-17.16-c1-0-9
Degree $2$
Conductor $2040$
Sign $0.242 - 0.970i$
Analytic cond. $16.2894$
Root an. cond. $4.03602$
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  − i·3-s + i·5-s − i·7-s − 9-s + 5i·11-s − 2·13-s + 15-s + (1 − 4i)17-s + 19-s − 21-s − 25-s + i·27-s + 7i·29-s + 5·33-s + 35-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.447i·5-s − 0.377i·7-s − 0.333·9-s + 1.50i·11-s − 0.554·13-s + 0.258·15-s + (0.242 − 0.970i)17-s + 0.229·19-s − 0.218·21-s − 0.200·25-s + 0.192i·27-s + 1.29i·29-s + 0.870·33-s + 0.169·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(16.2894\)
Root analytic conductor: \(4.03602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2040} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2040,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.193700872\)
\(L(\frac12)\) \(\approx\) \(1.193700872\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 - iT \)
17 \( 1 + (-1 + 4i)T \)
good7 \( 1 + iT - 7T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 7iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 5iT - 37T^{2} \)
41 \( 1 - 9iT - 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 - T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 - 11iT - 73T^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−9.436824266306539077115739299471, −8.411663382880398183010252755110, −7.38601109958429232862615806097, −7.18878085217130863602039753167, −6.39714027920381571725188022968, −5.18771117995872085159798708051, −4.57850760931319014195924351421, −3.30874356180755207220779417608, −2.41952786674362442059026864474, −1.31759160324951712304812672838, 0.43571308450040052551313773415, 2.03142965843514793218355429492, 3.23671207234235473722734977833, 3.96285336808496329123534989509, 5.02357387737898518869529491446, 5.72866231497059758665447994214, 6.35081967552229530508156187663, 7.64567750442555435173031308339, 8.340763559740114388860375945700, 8.946849259412356435270982116753

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