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Properties

Label 2-1638-91.81-c1-0-21
Degree $2$
Conductor $1638$
Sign $0.552 + 0.833i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.75 − 3.03i)5-s + (1.12 + 2.39i)7-s − 0.999·8-s − 3.50·10-s + 6.40·11-s + (−0.213 + 3.59i)13-s + (2.63 + 0.222i)14-s + (−0.5 + 0.866i)16-s + (2.33 + 4.05i)17-s + 5.23·19-s + (−1.75 + 3.03i)20-s + (3.20 − 5.54i)22-s + (−1.08 + 1.87i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.784 − 1.35i)5-s + (0.425 + 0.904i)7-s − 0.353·8-s − 1.10·10-s + 1.93·11-s + (−0.0590 + 0.998i)13-s + (0.704 + 0.0593i)14-s + (−0.125 + 0.216i)16-s + (0.567 + 0.982i)17-s + 1.20·19-s + (−0.392 + 0.679i)20-s + (0.683 − 1.18i)22-s + (−0.226 + 0.391i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.552 + 0.833i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.552 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.096871958\)
\(L(\frac12)\) \(\approx\) \(2.096871958\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-1.12 - 2.39i)T \)
13 \( 1 + (0.213 - 3.59i)T \)
good5 \( 1 + (1.75 + 3.03i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 6.40T + 11T^{2} \)
17 \( 1 + (-2.33 - 4.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 5.23T + 19T^{2} \)
23 \( 1 + (1.08 - 1.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.46 + 7.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.94 - 3.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.09 + 8.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.19 - 2.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.44 - 4.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.05 + 1.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.89 + 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 9.35T + 61T^{2} \)
67 \( 1 - 7.28T + 67T^{2} \)
71 \( 1 + (2.79 - 4.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.23 + 7.32i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.893 - 1.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.59T + 83T^{2} \)
89 \( 1 + (-3.50 + 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.92 - 8.53i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.288617664672834866012611740248, −8.615479803277991626045068206062, −7.973796523542100442215484812596, −6.71814884988443759271261557665, −5.77577856872778814805783123369, −4.94027807874224611793182475672, −4.14636289668335164700339177837, −3.53825876026106760450690451347, −1.84803938097210401635074452884, −1.10747891749745251526495632088, 0.988405058395318234240666943318, 3.00271036692506920273408367098, 3.57645286446673687982105978645, 4.40070715639992985467743357978, 5.45542931456740730405828974482, 6.70249915464998013070603729918, 6.92712498287281473506894057477, 7.70167988227920934393143321912, 8.391024778641377093941021865130, 9.572083481487341655603970315088

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