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Properties

Label 2-1638-13.10-c1-0-29
Degree $2$
Conductor $1638$
Sign $-0.582 + 0.812i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + 0.145i·5-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.0727 + 0.126i)10-s + (1.30 − 0.754i)11-s + (−2.95 − 2.06i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (0.160 − 0.277i)17-s + (−1.44 − 0.836i)19-s + (0.126 + 0.0727i)20-s + (0.754 − 1.30i)22-s + (−2.75 − 4.76i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.0650i·5-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (0.0230 + 0.0398i)10-s + (0.394 − 0.227i)11-s + (−0.819 − 0.572i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.0389 − 0.0674i)17-s + (−0.332 − 0.191i)19-s + (0.0281 + 0.0162i)20-s + (0.160 − 0.278i)22-s + (−0.573 − 0.994i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.582 + 0.812i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.582 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.778410894\)
\(L(\frac12)\) \(\approx\) \(1.778410894\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (2.95 + 2.06i)T \)
good5 \( 1 - 0.145iT - 5T^{2} \)
11 \( 1 + (-1.30 + 0.754i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.160 + 0.277i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.44 + 0.836i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.75 + 4.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.59 + 7.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.879iT - 31T^{2} \)
37 \( 1 + (-3.14 + 1.81i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.12 + 0.651i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.499 + 0.864i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.30iT - 47T^{2} \)
53 \( 1 + 5.12T + 53T^{2} \)
59 \( 1 + (2.78 + 1.60i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.63 + 2.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.99 + 1.15i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.10 + 3.52i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.35iT - 73T^{2} \)
79 \( 1 - 8.66T + 79T^{2} \)
83 \( 1 + 0.148iT - 83T^{2} \)
89 \( 1 + (7.13 - 4.11i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.4 - 6.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.305279966595070600744489041895, −8.283429266914118451137513742907, −7.40627501282829214277116416719, −6.53396067965912518345097450367, −5.83838170501996748855421882024, −4.82606198147207206914191030431, −4.05646366476631198041342882152, −3.03986824843620953415328037855, −2.15012273823028261366858999642, −0.53092144055006649249313659005, 1.68420540576754858263652638221, 2.87731892387512533803387941541, 3.85643664730134190881487047831, 4.73573300105811210921194337215, 5.55128771710179757524344377245, 6.45387441594350006652460122116, 7.13663496105902902029618354338, 7.87773254012855691956849798523, 8.945243909845750308115527395728, 9.485696362430670850992302757821

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