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Properties

Label 2-1040-260.167-c0-0-0
Degree $2$
Conductor $1040$
Sign $0.439 - 0.898i$
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  + (−0.866 + 0.5i)5-s + (0.866 + 0.5i)9-s + (−0.866 + 0.5i)13-s + (0.5 + 1.86i)17-s + (0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s + (0.5 + 0.866i)37-s + (−0.5 + 1.86i)41-s − 0.999·45-s + (−0.5 − 0.866i)49-s + (−0.366 + 0.366i)53-s + (0.866 − 1.5i)61-s + (0.499 − 0.866i)65-s − 1.73i·73-s + (0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)5-s + (0.866 + 0.5i)9-s + (−0.866 + 0.5i)13-s + (0.5 + 1.86i)17-s + (0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s + (0.5 + 0.866i)37-s + (−0.5 + 1.86i)41-s − 0.999·45-s + (−0.5 − 0.866i)49-s + (−0.366 + 0.366i)53-s + (0.866 − 1.5i)61-s + (0.499 − 0.866i)65-s − 1.73i·73-s + (0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.439 - 0.898i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :0),\ 0.439 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8764453874\)
\(L(\frac12)\) \(\approx\) \(0.8764453874\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
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 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
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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

−10.22681141113450834735899794860, −9.705930137150389091486198993003, −8.261498946932825119323805697809, −7.938158961598958426494252655727, −6.92747625142943793809926368742, −6.25539218111049845764065755931, −4.82598138099897767447104052016, −4.16993725920445531263329132758, −3.09855157384008797989438592043, −1.73702840638354076221423396016, 0.885628088825491717044332432951, 2.70386324440933465085607628320, 3.80614716672788373742730784941, 4.74494974616801830026428364823, 5.45990957445022399996624704970, 7.00582254908716858665105374959, 7.31655141796519093802545848902, 8.281407493489845666841573833690, 9.256225031678700513274046685856, 9.830584786746869741756966599166

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