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Properties

Label 2-1044-116.71-c0-0-0
Degree $2$
Conductor $1044$
Sign $-0.617 - 0.786i$
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  + (−0.433 + 0.900i)2-s + (−0.623 − 0.781i)4-s + (−1.75 − 0.846i)5-s + (0.974 − 0.222i)8-s + (1.52 − 1.21i)10-s + (−0.277 + 1.21i)13-s + (−0.222 + 0.974i)16-s + 1.80i·17-s + (0.433 + 1.90i)20-s + (1.74 + 2.19i)25-s + (−0.974 − 0.777i)26-s + (0.974 + 0.222i)29-s + (−0.781 − 0.623i)32-s + (−1.62 − 0.781i)34-s + (−1.52 + 0.347i)37-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (−0.623 − 0.781i)4-s + (−1.75 − 0.846i)5-s + (0.974 − 0.222i)8-s + (1.52 − 1.21i)10-s + (−0.277 + 1.21i)13-s + (−0.222 + 0.974i)16-s + 1.80i·17-s + (0.433 + 1.90i)20-s + (1.74 + 2.19i)25-s + (−0.974 − 0.777i)26-s + (0.974 + 0.222i)29-s + (−0.781 − 0.623i)32-s + (−1.62 − 0.781i)34-s + (−1.52 + 0.347i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $-0.617 - 0.786i$
Analytic conductor: \(0.521023\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :0),\ -0.617 - 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3985464888\)
\(L(\frac12)\) \(\approx\) \(0.3985464888\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.433 - 0.900i)T \)
3 \( 1 \)
29 \( 1 + (-0.974 - 0.222i)T \)
good5 \( 1 + (1.75 + 0.846i)T + (0.623 + 0.781i)T^{2} \)
7 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 - 1.80iT - T^{2} \)
19 \( 1 + (-0.222 + 0.974i)T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.623 + 0.781i)T^{2} \)
37 \( 1 + (1.52 - 0.347i)T + (0.900 - 0.433i)T^{2} \)
41 \( 1 - 1.24iT - T^{2} \)
43 \( 1 + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.781 - 0.376i)T + (0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.376 - 0.781i)T + (-0.623 + 0.781i)T^{2} \)
79 \( 1 + (-0.900 + 0.433i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.193 - 0.400i)T + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + (1.22 - 0.974i)T + (0.222 - 0.974i)T^{2} \)
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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

−10.31750429744824307161114159403, −9.233229825435400060620132072550, −8.447913315245517133812484111208, −8.155657631417271947639785498349, −7.15487041701843878312446518876, −6.46634540234348232194403254947, −5.16078776505263658411956292459, −4.38793485361499857743816678530, −3.71521180213701505724493159770, −1.42697028967573913078945143534, 0.46853900830064246834725029555, 2.70231240094035605598239285402, 3.25664156111365629198770287496, 4.24347166504844369120504206398, 5.18085911884763267002285793914, 6.95201537450331241507653524428, 7.47776876554304917579858463909, 8.165230961897785029905679827265, 8.981368244222250530686905339366, 10.11440808925501419215812119782

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