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Properties

Label 2-1044-116.51-c0-0-1
Degree $2$
Conductor $1044$
Sign $0.349 + 0.936i$
Analytic cond. $0.521023$
Root an. cond. $0.721819$
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  + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.347 − 1.52i)5-s + (0.781 − 0.623i)8-s + (−0.678 − 1.40i)10-s + (−1.12 + 1.40i)13-s + (0.623 − 0.781i)16-s − 0.445i·17-s + (−0.974 − 1.22i)20-s + (−1.30 + 0.626i)25-s + (−0.781 + 1.62i)26-s + (0.781 + 0.623i)29-s + (0.433 − 0.900i)32-s + (−0.0990 − 0.433i)34-s + (0.678 − 0.541i)37-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.347 − 1.52i)5-s + (0.781 − 0.623i)8-s + (−0.678 − 1.40i)10-s + (−1.12 + 1.40i)13-s + (0.623 − 0.781i)16-s − 0.445i·17-s + (−0.974 − 1.22i)20-s + (−1.30 + 0.626i)25-s + (−0.781 + 1.62i)26-s + (0.781 + 0.623i)29-s + (0.433 − 0.900i)32-s + (−0.0990 − 0.433i)34-s + (0.678 − 0.541i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $0.349 + 0.936i$
Analytic conductor: \(0.521023\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :0),\ 0.349 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.710220509\)
\(L(\frac12)\) \(\approx\) \(1.710220509\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.974 + 0.222i)T \)
3 \( 1 \)
29 \( 1 + (-0.781 - 0.623i)T \)
good5 \( 1 + (0.347 + 1.52i)T + (-0.900 + 0.433i)T^{2} \)
7 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + 0.445iT - T^{2} \)
19 \( 1 + (0.623 - 0.781i)T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.900 + 0.433i)T^{2} \)
37 \( 1 + (-0.678 + 0.541i)T + (0.222 - 0.974i)T^{2} \)
41 \( 1 - 1.80iT - T^{2} \)
43 \( 1 + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.222 - 0.974i)T^{2} \)
53 \( 1 + (0.433 + 1.90i)T + (-0.900 + 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.846 - 1.75i)T + (-0.623 - 0.781i)T^{2} \)
67 \( 1 + (0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-1.90 - 0.433i)T + (0.900 + 0.433i)T^{2} \)
79 \( 1 + (-0.222 + 0.974i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (1.21 - 0.277i)T + (0.900 - 0.433i)T^{2} \)
97 \( 1 + (0.376 + 0.781i)T + (-0.623 + 0.781i)T^{2} \)
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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

−9.834431668557275836982822310787, −9.302066298131165284928384281435, −8.276495829300791232279027759366, −7.33098846375533319147825961928, −6.47582278754217188732405485820, −5.29074508596784750388741762704, −4.67995724382273683678476513736, −4.07915957578496144031964426983, −2.62420663887542020562666385464, −1.35904443797505262076684347848, 2.40904022631874351370430049965, 3.06364718604947944603517630796, 4.00209191345660694751305716368, 5.15340458715805997395942189704, 6.06488850199766220937593387308, 6.85418502199538962752209053403, 7.59282655005251210284455522953, 8.156003057635019552538162276414, 9.764806084517464610459752913886, 10.63577189377436587012031778904

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