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Properties

Label 2-1044-29.4-c1-0-2
Degree $2$
Conductor $1044$
Sign $-0.372 - 0.927i$
Analytic cond. $8.33638$
Root an. cond. $2.88727$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.700 + 3.07i)5-s + (2.15 − 1.03i)7-s + (−2.67 + 2.13i)11-s + (2.48 + 3.11i)13-s − 0.723i·17-s + (−0.261 + 0.543i)19-s + (−0.402 − 1.76i)23-s + (−4.43 − 2.13i)25-s + (−0.849 + 5.31i)29-s + (−8.55 − 1.95i)31-s + (1.67 + 7.33i)35-s + (3.03 + 2.42i)37-s + 8.84i·41-s + (2.91 − 0.665i)43-s + (0.0107 − 0.00857i)47-s + ⋯
L(s)  = 1  + (−0.313 + 1.37i)5-s + (0.813 − 0.391i)7-s + (−0.807 + 0.643i)11-s + (0.688 + 0.862i)13-s − 0.175i·17-s + (−0.0600 + 0.124i)19-s + (−0.0839 − 0.367i)23-s + (−0.886 − 0.427i)25-s + (−0.157 + 0.987i)29-s + (−1.53 − 0.350i)31-s + (0.283 + 1.24i)35-s + (0.499 + 0.398i)37-s + 1.38i·41-s + (0.444 − 0.101i)43-s + (0.00156 − 0.00125i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $-0.372 - 0.927i$
Analytic conductor: \(8.33638\)
Root analytic conductor: \(2.88727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :1/2),\ -0.372 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.322085732\)
\(L(\frac12)\) \(\approx\) \(1.322085732\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 + (0.849 - 5.31i)T \)
good5 \( 1 + (0.700 - 3.07i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (-2.15 + 1.03i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (2.67 - 2.13i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (-2.48 - 3.11i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 0.723iT - 17T^{2} \)
19 \( 1 + (0.261 - 0.543i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (0.402 + 1.76i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (8.55 + 1.95i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (-3.03 - 2.42i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 - 8.84iT - 41T^{2} \)
43 \( 1 + (-2.91 + 0.665i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-0.0107 + 0.00857i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.486 + 2.13i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 5.86T + 59T^{2} \)
61 \( 1 + (-0.0283 - 0.0587i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (7.75 - 9.72i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-3.03 - 3.81i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-9.14 + 2.08i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + (4.06 + 3.24i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (-9.47 - 4.56i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (16.2 + 3.70i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (3.71 - 7.72i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.36790857789039476483996626070, −9.461665802958360760355710953347, −8.365758057068817269519171975495, −7.51503368760776138148385175895, −7.01234530298534979499841875653, −6.06646136473538128512156195883, −4.86648482691729108744878683239, −3.95826868794926507103004921295, −2.88993840799842219251194791171, −1.73998633518971957806523066546, 0.59984859501761902538546090261, 1.93726476581089618858904990939, 3.42926622873658380696458762211, 4.50940808364082444777513496126, 5.41001504668179544797819237497, 5.84869076198673925357806208845, 7.50363129488817283907635513211, 8.153536917574066733379894232396, 8.679144646520549041965767226512, 9.440237282161122289197006343256

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