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Properties

Label 2-1040-260.139-c0-0-0
Degree $2$
Conductor $1040$
Sign $0.711 - 0.702i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
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  + (0.5 + 0.866i)5-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)13-s + (−1.5 + 0.866i)17-s + (−0.499 + 0.866i)25-s + (0.5 − 0.866i)29-s + (1.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + (−0.499 + 0.866i)45-s + (0.5 − 0.866i)49-s − 1.73i·53-s + (−0.5 − 0.866i)61-s + 0.999·65-s − 1.73i·73-s + (−0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)13-s + (−1.5 + 0.866i)17-s + (−0.499 + 0.866i)25-s + (0.5 − 0.866i)29-s + (1.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + (−0.499 + 0.866i)45-s + (0.5 − 0.866i)49-s − 1.73i·53-s + (−0.5 − 0.866i)61-s + 0.999·65-s − 1.73i·73-s + (−0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :0),\ 0.711 - 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129305697\)
\(L(\frac12)\) \(\approx\) \(1.129305697\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.73iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−10.28558481728269334053421958089, −9.627721234195549596930256330891, −8.411844774688441449913490623094, −7.82595082795036048236814102944, −6.70641601798064300646658617272, −6.16730286968083165740175616918, −5.06111905997738891738270903112, −4.03041580921516312586752119241, −2.81418486962321400156465342698, −1.83744093531619581092117978911, 1.21034729290415664790833109033, 2.50881352554195027190935045500, 4.05709439824292527042181409115, 4.63786936763504828660811691709, 5.82433284345794124715512557434, 6.60547935327598181484981678761, 7.41135997117396243387991724267, 8.864461924519044392507222880360, 8.971212431242249039641714941042, 9.821019170323726796859061696046

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