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Properties

Label 2-1638-13.10-c1-0-14
Degree $2$
Conductor $1638$
Sign $0.902 - 0.431i$
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 + 0.732i·5-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (0.366 + 0.633i)10-s + (−1.5 + 0.866i)11-s + (1.59 + 3.23i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (−1.86 + 3.23i)17-s + (0.866 + 0.5i)19-s + (0.633 + 0.366i)20-s + (−0.866 + 1.5i)22-s + (1.73 + 3i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.327i·5-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (0.115 + 0.200i)10-s + (−0.452 + 0.261i)11-s + (0.443 + 0.896i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.452 + 0.783i)17-s + (0.198 + 0.114i)19-s + (0.141 + 0.0818i)20-s + (−0.184 + 0.319i)22-s + (0.361 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.902 - 0.431i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.902 - 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.447576786\)
\(L(\frac12)\) \(\approx\) \(2.447576786\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 - 3.23i)T \)
good5 \( 1 - 0.732iT - 5T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.86 - 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.23 - 5.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.19iT - 31T^{2} \)
37 \( 1 + (-5.83 + 3.36i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.63 + 2.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.46iT - 47T^{2} \)
53 \( 1 + 7T + 53T^{2} \)
59 \( 1 + (0.803 + 0.464i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.59 - 4.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.73 + 4.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.90 - 1.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.46iT - 73T^{2} \)
79 \( 1 - 2.07T + 79T^{2} \)
83 \( 1 - 0.196iT - 83T^{2} \)
89 \( 1 + (-9.06 + 5.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.5 - 7.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.462738210152191930777166342848, −8.748472342323068944538599892490, −7.77922849832951588386887083898, −6.86640724161462779695001214023, −6.17951509823235513325760044994, −5.20216383909316755572750853331, −4.42720917593724661239371599312, −3.48535681781101575635676774862, −2.45786563439892253109680955057, −1.42045467599086733677418857244, 0.824991909797718461465940940955, 2.50413742604742807317364036985, 3.35061895603064059981480144332, 4.59407574713558939662602346191, 5.05343742399378135050588325518, 6.04674905165356563452160751159, 6.81412061440473885144137882358, 7.79502036405850901701829937824, 8.324937270058030442492900953121, 9.167180307965718439152518910441

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