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Properties

Label 2-1638-13.4-c1-0-15
Degree $2$
Conductor $1638$
Sign $-0.612 - 0.790i$
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 + 3.71i·5-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (−1.85 + 3.21i)10-s + (5.00 + 2.88i)11-s + (2.87 − 2.17i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (0.106 + 0.183i)17-s + (1.85 − 1.06i)19-s + (−3.21 + 1.85i)20-s + (2.88 + 5.00i)22-s + (−1.23 + 2.14i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 1.65i·5-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (−0.586 + 1.01i)10-s + (1.50 + 0.870i)11-s + (0.798 − 0.602i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (0.0257 + 0.0445i)17-s + (0.424 − 0.245i)19-s + (−0.718 + 0.414i)20-s + (0.615 + 1.06i)22-s + (−0.258 + 0.447i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.612 - 0.790i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.612 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.563456621\)
\(L(\frac12)\) \(\approx\) \(2.563456621\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.87 + 2.17i)T \)
good5 \( 1 - 3.71iT - 5T^{2} \)
11 \( 1 + (-5.00 - 2.88i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.106 - 0.183i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.85 + 1.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.23 - 2.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0492 - 0.0853i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.31iT - 31T^{2} \)
37 \( 1 + (-6.81 - 3.93i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.51 + 3.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.28 - 3.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.15iT - 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + (-0.200 + 0.115i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.01 + 6.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.2 + 6.48i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.37 - 3.68i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.60iT - 73T^{2} \)
79 \( 1 + 9.19T + 79T^{2} \)
83 \( 1 + 3.17iT - 83T^{2} \)
89 \( 1 + (-10.2 - 5.90i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.1 + 7.04i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.773157146297101680109322819342, −8.865734811432131250014358099705, −7.74319792409388770065728540636, −7.06519065642626608370977252666, −6.41736874539119322601006924247, −5.92555706759219793445398617515, −4.61320847282357815519772962784, −3.55735233037312446326891956200, −3.09752398148275054220997491565, −1.79492925072440445174809807495, 0.867642148442995386340932855182, 1.63691278720637472048026839068, 3.31106223181897185569106254609, 4.15380511798108468171866969362, 4.69078700147954747819462794069, 5.98831891808345238221922356505, 6.18662494714937627388306525090, 7.54114135018657498134688094216, 8.597655397426621352890477173610, 9.090818814422220670430474699483

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