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Properties

Label 2-2040-17.4-c1-0-13
Degree $2$
Conductor $2040$
Sign $0.999 + 0.00120i$
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 + (0.852 + 0.852i)7-s − 1.00i·9-s + (−3.47 − 3.47i)11-s + 2.90·13-s − 1.00i·15-s + (−2.54 + 3.24i)17-s − 4.79i·19-s − 1.20·21-s + (4.27 + 4.27i)23-s − 1.00i·25-s + (0.707 + 0.707i)27-s + (1.54 − 1.54i)29-s + (3.86 − 3.86i)31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.316 + 0.316i)5-s + (0.322 + 0.322i)7-s − 0.333i·9-s + (−1.04 − 1.04i)11-s + 0.805·13-s − 0.258i·15-s + (−0.616 + 0.787i)17-s − 1.10i·19-s − 0.263·21-s + (0.890 + 0.890i)23-s − 0.200i·25-s + (0.136 + 0.136i)27-s + (0.287 − 0.287i)29-s + (0.694 − 0.694i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.999 + 0.00120i$
Analytic conductor: \(16.2894\)
Root analytic conductor: \(4.03602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2040} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2040,\ (\ :1/2),\ 0.999 + 0.00120i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.306534603\)
\(L(\frac12)\) \(\approx\) \(1.306534603\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 - 3.24i)T \)
good7 \( 1 + (-0.852 - 0.852i)T + 7iT^{2} \)
11 \( 1 + (3.47 + 3.47i)T + 11iT^{2} \)
13 \( 1 - 2.90T + 13T^{2} \)
19 \( 1 + 4.79iT - 19T^{2} \)
23 \( 1 + (-4.27 - 4.27i)T + 23iT^{2} \)
29 \( 1 + (-1.54 + 1.54i)T - 29iT^{2} \)
31 \( 1 + (-3.86 + 3.86i)T - 31iT^{2} \)
37 \( 1 + (0.672 - 0.672i)T - 37iT^{2} \)
41 \( 1 + (0.644 + 0.644i)T + 41iT^{2} \)
43 \( 1 + 12.6iT - 43T^{2} \)
47 \( 1 - 5.78T + 47T^{2} \)
53 \( 1 - 3.33iT - 53T^{2} \)
59 \( 1 - 4.62iT - 59T^{2} \)
61 \( 1 + (-4.42 - 4.42i)T + 61iT^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + (1.68 - 1.68i)T - 71iT^{2} \)
73 \( 1 + (10.8 - 10.8i)T - 73iT^{2} \)
79 \( 1 + (-2.13 - 2.13i)T + 79iT^{2} \)
83 \( 1 - 3.95iT - 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (-6.53 + 6.53i)T - 97iT^{2} \)
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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

−8.813090927882132349738183663157, −8.633442540855282582061523254146, −7.59592097642749815490648454477, −6.72058415157713425097140434128, −5.81936344510753912485766666698, −5.25273852644293143573480703646, −4.21730044293601841125921934172, −3.33969303044431090129809826092, −2.36171628963697560913854043172, −0.67738710298752653399302791867, 0.890360657392568458310097056627, 2.07486226854669870464052202785, 3.23203059292148376355326933167, 4.60994823578873839054185765169, 4.86761389183899931522944237326, 6.03716364818293827152793269283, 6.84483091645369598285549774479, 7.61660536299167257276358248891, 8.197300473518544382780532131378, 9.030439141781305382918937192519

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