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Properties

Label 2-1044-29.13-c1-0-6
Degree $2$
Conductor $1044$
Sign $0.992 - 0.122i$
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  + (1.25 + 0.606i)5-s + (1.39 − 1.75i)7-s + (4.39 + 1.00i)11-s + (−0.444 + 1.94i)13-s + 2.28i·17-s + (−2.02 + 1.61i)19-s + (2.43 − 1.17i)23-s + (−1.89 − 2.38i)25-s + (3.73 − 3.87i)29-s + (0.190 − 0.396i)31-s + (2.82 − 1.35i)35-s + (−0.614 + 0.140i)37-s − 1.00i·41-s + (−0.679 − 1.41i)43-s + (7.74 + 1.76i)47-s + ⋯
L(s)  = 1  + (0.563 + 0.271i)5-s + (0.527 − 0.661i)7-s + (1.32 + 0.302i)11-s + (−0.123 + 0.539i)13-s + 0.554i·17-s + (−0.463 + 0.369i)19-s + (0.507 − 0.244i)23-s + (−0.379 − 0.476i)25-s + (0.693 − 0.720i)29-s + (0.0342 − 0.0711i)31-s + (0.476 − 0.229i)35-s + (−0.100 + 0.0230i)37-s − 0.157i·41-s + (−0.103 − 0.215i)43-s + (1.12 + 0.257i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $0.992 - 0.122i$
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: \((2,\ 1044,\ (\ :1/2),\ 0.992 - 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.022548286\)
\(L(\frac12)\) \(\approx\) \(2.022548286\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.73 + 3.87i)T \)
good5 \( 1 + (-1.25 - 0.606i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-1.39 + 1.75i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (-4.39 - 1.00i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.444 - 1.94i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 2.28iT - 17T^{2} \)
19 \( 1 + (2.02 - 1.61i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (-2.43 + 1.17i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (-0.190 + 0.396i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (0.614 - 0.140i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 + 1.00iT - 41T^{2} \)
43 \( 1 + (0.679 + 1.41i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-7.74 - 1.76i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (-3.60 - 1.73i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 - 8.70T + 59T^{2} \)
61 \( 1 + (-1.15 - 0.920i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (-1.36 - 5.99i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (3.27 - 14.3i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (7.07 + 14.6i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (5.25 - 1.19i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (-3.64 - 4.57i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-4.21 + 8.75i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (-1.25 + 1.00i)T + (21.5 - 94.5i)T^{2} \)
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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

−10.03810861233269243502296427298, −9.128889983312537328713945589470, −8.365827120339805479406923853869, −7.29988630350229131753682325633, −6.57973044187905775434819508723, −5.82080604480762462498509674349, −4.46351677273603351481200380642, −3.92855749444735892585129491633, −2.36161729288908978774376792724, −1.27615604146847390523785447327, 1.18351104800811856667398012438, 2.38737754644608117838053911362, 3.61454496681758491028826460090, 4.84275714800653062985987678446, 5.55582696774868328460633668239, 6.46155143009772297249728524373, 7.35030188592208735856410258746, 8.535952752731628290000405254480, 8.989344003965804186541096784422, 9.757222558825899421965353133505

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