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Properties

Label 2-1040-260.227-c0-0-0
Degree $2$
Conductor $1040$
Sign $0.507 - 0.861i$
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 + (−0.866 + 0.5i)9-s + (−0.5 + 0.866i)13-s + (1.86 + 0.5i)17-s + (−0.499 + 0.866i)25-s + (−0.866 − 0.5i)29-s + (0.866 + 0.5i)37-s + (−0.5 − 1.86i)41-s + (−0.866 − 0.499i)45-s + (0.5 − 0.866i)49-s + (−0.366 + 0.366i)53-s + (0.866 + 1.5i)61-s − 0.999·65-s + 1.73·73-s + (0.499 − 0.866i)81-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (−0.866 + 0.5i)9-s + (−0.5 + 0.866i)13-s + (1.86 + 0.5i)17-s + (−0.499 + 0.866i)25-s + (−0.866 − 0.5i)29-s + (0.866 + 0.5i)37-s + (−0.5 − 1.86i)41-s + (−0.866 − 0.499i)45-s + (0.5 − 0.866i)49-s + (−0.366 + 0.366i)53-s + (0.866 + 1.5i)61-s − 0.999·65-s + 1.73·73-s + (0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.507 - 0.861i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :0),\ 0.507 - 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.038584219\)
\(L(\frac12)\) \(\approx\) \(1.038584219\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 - 1.73T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.17886312555609141892251341590, −9.632058423815120778660360073225, −8.602090212042757684029898603999, −7.68308333513940708176272342699, −6.96910468230477072697984530100, −5.87203907593303928734825381093, −5.37805185668941678787284967365, −3.93615405735601290505262500817, −2.91229937202761209843849778530, −1.90793440559899135848859458564, 1.04014313862407124036476588475, 2.65139280817255736017095419241, 3.64024947934814961040560133451, 5.09311792111501455382155942083, 5.51394511352485799079711138842, 6.44305784828479824613403495995, 7.78392061831040915221221870002, 8.216377242418838230786180249640, 9.487713022419574572661661866185, 9.603294767757080253540090424679

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