↑

Properties

Label 2-1638-13.4-c1-0-29
Degree $2$
Conductor $1638$
Sign $-0.994 + 0.107i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + 1.34i·5-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (0.674 − 1.16i)10-s + (−1.11 − 0.646i)11-s + (−0.343 − 3.58i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (−1.76 − 3.05i)17-s + (1.98 − 1.14i)19-s + (−1.16 + 0.674i)20-s + (0.646 + 1.11i)22-s + (−3.08 + 5.33i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.603i·5-s + (−0.327 + 0.188i)7-s − 0.353i·8-s + (0.213 − 0.369i)10-s + (−0.337 − 0.194i)11-s + (−0.0951 − 0.995i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (−0.428 − 0.741i)17-s + (0.456 − 0.263i)19-s + (−0.261 + 0.150i)20-s + (0.137 + 0.238i)22-s + (−0.642 + 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.994 + 0.107i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.994 + 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1031680819\)
\(L(\frac12)\) \(\approx\) \(0.1031680819\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.343 + 3.58i)T \)
good5 \( 1 - 1.34iT - 5T^{2} \)
11 \( 1 + (1.11 + 0.646i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.76 + 3.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.98 + 1.14i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.08 - 5.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.11 - 7.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.90iT - 31T^{2} \)
37 \( 1 + (-3.71 - 2.14i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.84 + 3.95i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.26 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.7iT - 47T^{2} \)
53 \( 1 + 3.30T + 53T^{2} \)
59 \( 1 + (9.40 - 5.42i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.64 + 4.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.85 + 4.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.89 - 3.97i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.28iT - 73T^{2} \)
79 \( 1 + 3.32T + 79T^{2} \)
83 \( 1 - 0.731iT - 83T^{2} \)
89 \( 1 + (11.5 + 6.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.5 - 6.10i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.974137389666334159561128415139, −8.395712390915394339710320119843, −7.23534577705084033291102433105, −7.00216928407997907552079343977, −5.71530261455199424245492850161, −4.96847401870982487563128773714, −3.33724927298120348770749567411, −3.05274677017572092848721441345, −1.68608773924625116248925494309, −0.04768929803600356780339510548, 1.47910673539787790050894664144, 2.60385379953403535529935889017, 4.13040400230992499051869271770, 4.73955566126573281467861851815, 6.06297760068084043979393342559, 6.41948096746628950208931450290, 7.63473897658579288124389179941, 8.084512847265014830087298529838, 9.053969482839738702009756691371, 9.603973993377395487958106795555

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