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Properties

Label 2-1040-65.64-c1-0-4
Degree $2$
Conductor $1040$
Sign $-0.339 - 0.940i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + 0.949i·3-s + (0.108 − 2.23i)5-s − 3.85·7-s + 2.09·9-s + 0.589i·11-s + (−3.32 + 1.38i)13-s + (2.11 + 0.103i)15-s + 3.66i·17-s + 5.94i·19-s − 3.66i·21-s − 3.51i·23-s + (−4.97 − 0.486i)25-s + 4.83i·27-s + 5.33·29-s + 8.71i·31-s + ⋯
L(s)  = 1  + 0.547i·3-s + (0.0486 − 0.998i)5-s − 1.45·7-s + 0.699·9-s + 0.177i·11-s + (−0.922 + 0.384i)13-s + (0.547 + 0.0266i)15-s + 0.887i·17-s + 1.36i·19-s − 0.798i·21-s − 0.733i·23-s + (−0.995 − 0.0972i)25-s + 0.931i·27-s + 0.991·29-s + 1.56i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.339 - 0.940i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.339 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8823278467\)
\(L(\frac12)\) \(\approx\) \(0.8823278467\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.108 + 2.23i)T \)
13 \( 1 + (3.32 - 1.38i)T \)
good3 \( 1 - 0.949iT - 3T^{2} \)
7 \( 1 + 3.85T + 7T^{2} \)
11 \( 1 - 0.589iT - 11T^{2} \)
17 \( 1 - 3.66iT - 17T^{2} \)
19 \( 1 - 5.94iT - 19T^{2} \)
23 \( 1 + 3.51iT - 23T^{2} \)
29 \( 1 - 5.33T + 29T^{2} \)
31 \( 1 - 8.71iT - 31T^{2} \)
37 \( 1 - 1.85T + 37T^{2} \)
41 \( 1 - 4.63iT - 41T^{2} \)
43 \( 1 - 6.30iT - 43T^{2} \)
47 \( 1 - 3.85T + 47T^{2} \)
53 \( 1 - 10.7iT - 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 - 4.09T + 67T^{2} \)
71 \( 1 + 5.34iT - 71T^{2} \)
73 \( 1 + 6.09T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 8.77T + 83T^{2} \)
89 \( 1 - 0.413iT - 89T^{2} \)
97 \( 1 - 8.45T + 97T^{2} \)
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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

−10.01987377992825993105191139438, −9.518998709466799663700679183487, −8.723927613163514182077525828193, −7.75221658880844510433329997088, −6.69172118808222146289934404819, −5.94694632144517142509705272891, −4.76119924687355093022434679336, −4.14496391015551189828772385887, −3.05904735142757547288927705272, −1.48415001492905069308488050465, 0.39863465835384188878208771331, 2.41305728711922068985552708235, 3.05128006368634466300430933463, 4.24329497149181375798588008985, 5.58860218663814591290770428242, 6.53773534733821631883608813015, 7.12695818047644909567328049635, 7.58947259533143906666943863399, 9.048946604252813058502356459140, 9.838953161255959451919229488945

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