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Properties

Label 2-1044-29.13-c1-0-3
Degree $2$
Conductor $1044$
Sign $-0.561 - 0.827i$
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  + (2.10 + 1.01i)5-s + (−2.51 + 3.15i)7-s + (−1.80 − 0.411i)11-s + (0.177 − 0.778i)13-s + 6.12i·17-s + (0.496 − 0.396i)19-s + (−5.06 + 2.44i)23-s + (0.274 + 0.344i)25-s + (−5.15 − 1.55i)29-s + (−0.776 + 1.61i)31-s + (−8.49 + 4.08i)35-s + (6.78 − 1.54i)37-s + 10.0i·41-s + (−1.73 − 3.59i)43-s + (−7.60 − 1.73i)47-s + ⋯
L(s)  = 1  + (0.939 + 0.452i)5-s + (−0.952 + 1.19i)7-s + (−0.543 − 0.124i)11-s + (0.0492 − 0.215i)13-s + 1.48i·17-s + (0.113 − 0.0909i)19-s + (−1.05 + 0.509i)23-s + (0.0549 + 0.0689i)25-s + (−0.957 − 0.288i)29-s + (−0.139 + 0.289i)31-s + (−1.43 + 0.691i)35-s + (1.11 − 0.254i)37-s + 1.56i·41-s + (−0.264 − 0.548i)43-s + (−1.10 − 0.253i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $-0.561 - 0.827i$
Analytic conductor: \(8.33638\)
Root analytic conductor: \(2.88727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :1/2),\ -0.561 - 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.162129569\)
\(L(\frac12)\) \(\approx\) \(1.162129569\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.15 + 1.55i)T \)
good5 \( 1 + (-2.10 - 1.01i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (2.51 - 3.15i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (1.80 + 0.411i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.177 + 0.778i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 6.12iT - 17T^{2} \)
19 \( 1 + (-0.496 + 0.396i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (5.06 - 2.44i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (0.776 - 1.61i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (-6.78 + 1.54i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 - 10.0iT - 41T^{2} \)
43 \( 1 + (1.73 + 3.59i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (7.60 + 1.73i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (-7.82 - 3.76i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 - 0.770T + 59T^{2} \)
61 \( 1 + (2.48 + 1.98i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (0.719 + 3.15i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (2.20 - 9.67i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.21 - 10.8i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (-8.51 + 1.94i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (-4.83 - 6.06i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (0.850 - 1.76i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (4.74 - 3.78i)T + (21.5 - 94.5i)T^{2} \)
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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

−9.951383522710456820493216830351, −9.632295285214652050016500240782, −8.607256101586504376916188721155, −7.82590997360874965871819096729, −6.50099606544201222707795206275, −5.98587193178111407097483760385, −5.40202579812701930475673353769, −3.83177862635726478310005240295, −2.77066847432877822154006284149, −1.93454615858670981220848037904, 0.48697003224786041426651192873, 2.04072783007819870681513983399, 3.28176967376436036754967172591, 4.36716621863932133433928697222, 5.35484107169657371338949720683, 6.24136812155541798371269294661, 7.11246751539715241522300043354, 7.83069141836891955576836058377, 9.111454393498960495961331608044, 9.689743482276238704490031050554

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