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Properties

Label 2-2040-17.13-c1-0-23
Degree $2$
Conductor $2040$
Sign $0.887 - 0.460i$
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  + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−1.24 + 1.24i)7-s + 1.00i·9-s + (2.10 − 2.10i)11-s + 3.64·13-s + 1.00i·15-s + (3.74 + 1.71i)17-s − 8.09i·19-s − 1.75·21-s + (5.38 − 5.38i)23-s + 1.00i·25-s + (−0.707 + 0.707i)27-s + (−6.04 − 6.04i)29-s + (3.84 + 3.84i)31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.316 + 0.316i)5-s + (−0.469 + 0.469i)7-s + 0.333i·9-s + (0.635 − 0.635i)11-s + 1.01·13-s + 0.258i·15-s + (0.909 + 0.416i)17-s − 1.85i·19-s − 0.383·21-s + (1.12 − 1.12i)23-s + 0.200i·25-s + (−0.136 + 0.136i)27-s + (−1.12 − 1.12i)29-s + (0.690 + 0.690i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.887 - 0.460i$
Analytic conductor: \(16.2894\)
Root analytic conductor: \(4.03602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2040} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2040,\ (\ :1/2),\ 0.887 - 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.340235026\)
\(L(\frac12)\) \(\approx\) \(2.340235026\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (-3.74 - 1.71i)T \)
good7 \( 1 + (1.24 - 1.24i)T - 7iT^{2} \)
11 \( 1 + (-2.10 + 2.10i)T - 11iT^{2} \)
13 \( 1 - 3.64T + 13T^{2} \)
19 \( 1 + 8.09iT - 19T^{2} \)
23 \( 1 + (-5.38 + 5.38i)T - 23iT^{2} \)
29 \( 1 + (6.04 + 6.04i)T + 29iT^{2} \)
31 \( 1 + (-3.84 - 3.84i)T + 31iT^{2} \)
37 \( 1 + (-2.27 - 2.27i)T + 37iT^{2} \)
41 \( 1 + (6.86 - 6.86i)T - 41iT^{2} \)
43 \( 1 - 1.46iT - 43T^{2} \)
47 \( 1 - 7.37T + 47T^{2} \)
53 \( 1 + 0.895iT - 53T^{2} \)
59 \( 1 - 10.7iT - 59T^{2} \)
61 \( 1 + (1.89 - 1.89i)T - 61iT^{2} \)
67 \( 1 + 0.472T + 67T^{2} \)
71 \( 1 + (-9.42 - 9.42i)T + 71iT^{2} \)
73 \( 1 + (2.88 + 2.88i)T + 73iT^{2} \)
79 \( 1 + (-7.18 + 7.18i)T - 79iT^{2} \)
83 \( 1 + 2.60iT - 83T^{2} \)
89 \( 1 - 8.81T + 89T^{2} \)
97 \( 1 + (4.63 + 4.63i)T + 97iT^{2} \)
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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

−8.991921129833209404721522259092, −8.769835214321513805213935162536, −7.73609830541108986343249246460, −6.63521390771310791314989233906, −6.17241120460042060659941903253, −5.20150868289626596338169438778, −4.18869132490846046244704971107, −3.20550390948345959311164460871, −2.61207049086570413903872099966, −1.06325069845614026375663822178, 1.09120661468607169593702558964, 1.89772006898133662175698825187, 3.48577413376624588033443180164, 3.76472131529202672024345511172, 5.20500317167007371515346138820, 5.93601799898215373268412232096, 6.83227566649054197118768743925, 7.50160235243523165874218513077, 8.263313755443188315329635468652, 9.193351432043721790825581686487

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