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Properties

Label 2-1044-29.22-c1-0-5
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: $\chi_{1044} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :1/2),\ 0.372 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.957262024\)
\(L(\frac12)\) \(\approx\) \(1.957262024\)
\(L(\frac{3}{2})\) 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

−10.18978066910845555829626435029, −9.295746726680746749135113439246, −8.415472914058388652395696016988, −7.45990768298003080930100937886, −6.74031925299068518998336331289, −5.91937854141494223844568046841, −4.93990419076177530099507768432, −3.70463724015114915482073676355, −2.73645048258734978021665562682, −1.59859221532239566093095613269, 1.00887054948208095617104377939, 1.88439505973347083737258366282, 3.80079619864325822048076272567, 4.42295847314427843582652542035, 5.45135891792196123577270205197, 6.19466093881731150079022602918, 7.37575983579803824613356899172, 8.310177082613479246438472390562, 8.946927134607068793753637672561, 9.468245048912643450909918320677

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