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Properties

Label 2-1638-91.4-c1-0-1
Degree $2$
Conductor $1638$
Sign $-0.620 - 0.784i$
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  − i·2-s − 4-s + (−0.552 + 0.319i)5-s + (2.23 + 1.41i)7-s + i·8-s + (0.319 + 0.552i)10-s + (−3.93 + 2.27i)11-s + (2.48 + 2.61i)13-s + (1.41 − 2.23i)14-s + 16-s + 0.708·17-s + (−6.36 − 3.67i)19-s + (0.552 − 0.319i)20-s + (2.27 + 3.93i)22-s − 6.50·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.247 + 0.142i)5-s + (0.845 + 0.533i)7-s + 0.353i·8-s + (0.100 + 0.174i)10-s + (−1.18 + 0.685i)11-s + (0.688 + 0.725i)13-s + (0.377 − 0.598i)14-s + 0.250·16-s + 0.171·17-s + (−1.45 − 0.842i)19-s + (0.123 − 0.0713i)20-s + (0.484 + 0.839i)22-s − 1.35·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.620 - 0.784i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.620 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3185102201\)
\(L(\frac12)\) \(\approx\) \(0.3185102201\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (-2.23 - 1.41i)T \)
13 \( 1 + (-2.48 - 2.61i)T \)
good5 \( 1 + (0.552 - 0.319i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.93 - 2.27i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.708T + 17T^{2} \)
19 \( 1 + (6.36 + 3.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.50T + 23T^{2} \)
29 \( 1 + (-3.63 + 6.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.75 + 3.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.10iT - 37T^{2} \)
41 \( 1 + (4.62 + 2.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.21 + 9.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.68 - 0.972i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.20 - 5.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.31iT - 59T^{2} \)
61 \( 1 + (3.41 - 5.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.39 - 1.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.11 + 5.26i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.55 + 3.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.48 - 9.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.41iT - 83T^{2} \)
89 \( 1 + 5.53iT - 89T^{2} \)
97 \( 1 + (7.96 - 4.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.753639477914224930899740878747, −8.857928492090648981711191250380, −8.196273181139965276246577209444, −7.52027277350452284799812960056, −6.35828040295261267109462476799, −5.39805752490880715907585153979, −4.55538648588359323049079280224, −3.81082065031127138877071694551, −2.40306629332889969875805600230, −1.85594630005477798986952052556, 0.11563147826313511124293894708, 1.69419708055631348554551153832, 3.26419747586601218982386136937, 4.18388740876278212005043090858, 5.07548928357917494852033721408, 5.84075215503559828403724585314, 6.64015859115438033369400546936, 7.84250618820277254545646102935, 8.152912981084294962666155695663, 8.615219565814791059983237428273

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