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Properties

Label 2-1638-273.68-c1-0-32
Degree $2$
Conductor $1638$
Sign $-0.682 + 0.730i$
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 + (0.831 − 1.44i)5-s + (−0.654 − 2.56i)7-s + i·8-s + (−1.44 − 0.831i)10-s + (0.423 + 0.244i)11-s + (2.73 + 2.35i)13-s + (−2.56 + 0.654i)14-s + 16-s + 3.32·17-s + (5.15 − 2.97i)19-s + (−0.831 + 1.44i)20-s + (0.244 − 0.423i)22-s − 7.51i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.372 − 0.644i)5-s + (−0.247 − 0.968i)7-s + 0.353i·8-s + (−0.455 − 0.263i)10-s + (0.127 + 0.0736i)11-s + (0.757 + 0.652i)13-s + (−0.685 + 0.174i)14-s + 0.250·16-s + 0.807·17-s + (1.18 − 0.682i)19-s + (−0.186 + 0.322i)20-s + (0.0521 − 0.0902i)22-s − 1.56i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.682 + 0.730i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.682 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.699381808\)
\(L(\frac12)\) \(\approx\) \(1.699381808\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.654 + 2.56i)T \)
13 \( 1 + (-2.73 - 2.35i)T \)
good5 \( 1 + (-0.831 + 1.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.423 - 0.244i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.32T + 17T^{2} \)
19 \( 1 + (-5.15 + 2.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.51iT - 23T^{2} \)
29 \( 1 + (-1.89 + 1.09i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.79 - 2.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.02T + 37T^{2} \)
41 \( 1 + (2.04 + 3.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.67 + 2.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.89 + 8.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.998 - 0.576i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.693T + 59T^{2} \)
61 \( 1 + (5.82 - 3.36i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.57 - 4.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.02 + 2.32i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.99 - 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.22 + 3.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.84T + 83T^{2} \)
89 \( 1 + 5.05T + 89T^{2} \)
97 \( 1 + (5.21 + 3.01i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.027868884782847693891212167182, −8.696086864784896749869192443051, −7.46482797492356237278305209309, −6.76852642510812921732315393395, −5.63195725179008019484172254262, −4.80444075874906913489340563450, −3.93659653160215986033748011206, −3.08313279274970696703091657908, −1.66521077824242601388883357379, −0.73070178169911439998153362888, 1.43687004577961245368131828977, 2.98710414598934880044029198175, 3.57689486457852146845350058980, 5.10684150896999706754254799852, 5.79309047217087779632661345798, 6.23803799092315442408297503888, 7.35308605505505691415408943166, 7.947951867652162408769200474369, 8.872747039491329879476538757999, 9.582904768541591760565086907327

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