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Properties

Label 2-1044-116.63-c0-0-0
Degree $2$
Conductor $1044$
Sign $0.303 - 0.952i$
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  + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (0.541 + 0.678i)5-s + (0.433 + 0.900i)8-s + (−0.846 − 0.193i)10-s + (0.400 + 0.193i)13-s + (−0.900 − 0.433i)16-s + 1.24i·17-s + (0.781 − 0.376i)20-s + (0.0549 − 0.240i)25-s + (−0.433 + 0.0990i)26-s + (0.433 − 0.900i)29-s + (0.974 − 0.222i)32-s + (−0.777 − 0.974i)34-s + (0.846 + 1.75i)37-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (0.541 + 0.678i)5-s + (0.433 + 0.900i)8-s + (−0.846 − 0.193i)10-s + (0.400 + 0.193i)13-s + (−0.900 − 0.433i)16-s + 1.24i·17-s + (0.781 − 0.376i)20-s + (0.0549 − 0.240i)25-s + (−0.433 + 0.0990i)26-s + (0.433 − 0.900i)29-s + (0.974 − 0.222i)32-s + (−0.777 − 0.974i)34-s + (0.846 + 1.75i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $0.303 - 0.952i$
Analytic conductor: \(0.521023\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :0),\ 0.303 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7859607717\)
\(L(\frac12)\) \(\approx\) \(0.7859607717\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 - 0.623i)T \)
3 \( 1 \)
29 \( 1 + (-0.433 + 0.900i)T \)
good5 \( 1 + (-0.541 - 0.678i)T + (-0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 - 1.24iT - T^{2} \)
19 \( 1 + (-0.900 - 0.433i)T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.222 + 0.974i)T^{2} \)
37 \( 1 + (-0.846 - 1.75i)T + (-0.623 + 0.781i)T^{2} \)
41 \( 1 - 0.445iT - T^{2} \)
43 \( 1 + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.974 + 1.22i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-1.52 + 0.347i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-1.22 - 0.974i)T + (0.222 + 0.974i)T^{2} \)
79 \( 1 + (0.623 - 0.781i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (1.40 - 1.12i)T + (0.222 - 0.974i)T^{2} \)
97 \( 1 + (1.90 + 0.433i)T + (0.900 + 0.433i)T^{2} \)
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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

−9.962982280314149718198851472101, −9.693816168380471870223281448343, −8.361032702107533412980848789069, −8.074700420212063532129312714299, −6.65995571961707738837622994995, −6.43356927311386914308105447728, −5.45859911879075299934087798481, −4.25331516126461804658913001481, −2.75466688350496495892879662539, −1.55965360603668715718980070702, 1.07842876368033985886516891147, 2.33742522378031155051489215453, 3.44910634637365981737457304547, 4.64557388503985067329503075895, 5.62262509565858132729546085628, 6.81559742901035137610841394812, 7.63287731103127842220588535890, 8.572099327399595091182322518353, 9.248322822734600210399035302419, 9.753999442401643058959162592559

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