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Properties

Label 2-1638-91.51-c1-0-36
Degree $2$
Conductor $1638$
Sign $-0.207 + 0.978i$
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 + (1.95 + 1.13i)5-s + (−2.41 + 1.07i)7-s − 0.999i·8-s + (−1.13 − 1.95i)10-s + (1.09 − 0.630i)11-s + (−1.26 − 3.37i)13-s + (2.63 + 0.279i)14-s + (−0.5 + 0.866i)16-s + (0.188 + 0.326i)17-s + (−6.71 − 3.87i)19-s + 2.26i·20-s − 1.26·22-s + (2.24 − 3.89i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.875 + 0.505i)5-s + (−0.914 + 0.405i)7-s − 0.353i·8-s + (−0.357 − 0.619i)10-s + (0.329 − 0.190i)11-s + (−0.349 − 0.936i)13-s + (0.703 + 0.0748i)14-s + (−0.125 + 0.216i)16-s + (0.0457 + 0.0793i)17-s + (−1.54 − 0.889i)19-s + 0.505i·20-s − 0.269·22-s + (0.468 − 0.811i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.207 + 0.978i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.207 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8728450319\)
\(L(\frac12)\) \(\approx\) \(0.8728450319\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 - 1.07i)T \)
13 \( 1 + (1.26 + 3.37i)T \)
good5 \( 1 + (-1.95 - 1.13i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.09 + 0.630i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.188 - 0.326i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.71 + 3.87i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.24 + 3.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.40T + 29T^{2} \)
31 \( 1 + (3.46 - 2i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.92 + 1.68i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.90iT - 41T^{2} \)
43 \( 1 - 8.75T + 43T^{2} \)
47 \( 1 + (2.85 + 1.64i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.04 - 2.33i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.45 + 7.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.11 - 0.646i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.11iT - 71T^{2} \)
73 \( 1 + (-8.26 + 4.77i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.14 + 5.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.39iT - 83T^{2} \)
89 \( 1 + (-9.85 - 5.68i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.03iT - 97T^{2} \)
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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

−9.018269895810445124943729886385, −8.803936947986876306021286653532, −7.57024453396623718808656201541, −6.62767773083053890342599146444, −6.20256642333111540706955209150, −5.15846967124020837080763104965, −3.81142186457375911156132694478, −2.75365407798414527294600026049, −2.17889050012532659263434840536, −0.41061199860753755459779735739, 1.31827484476175414806389117437, 2.30494074914935072704231293172, 3.74142982930473807020994963817, 4.73307393604570195784228152989, 5.82665025016399285941760946707, 6.42955135744376054505967605651, 7.12049370977208877820228070726, 8.087208378773448750345630894588, 9.075448093925616350560523015954, 9.465990974831268794184845667384

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