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Properties

Label 2-1638-13.10-c1-0-0
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

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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: $\chi_{1638} (127, \cdot )$
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 \( 1 + (10.5 + 6.10i)T + (48.5 + 84.0i)T^{2} \)
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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

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

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