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Properties

Label 2-1638-13.4-c1-0-12
Degree $2$
Conductor $1638$
Sign $-0.0453 - 0.998i$
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 + 1.24i·5-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (−0.623 + 1.08i)10-s + (−1.41 − 0.816i)11-s + (3.60 − 0.117i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (2.78 + 4.81i)17-s + (−4.23 + 2.44i)19-s + (−1.08 + 0.623i)20-s + (−0.816 − 1.41i)22-s + (−2.83 + 4.91i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 0.557i·5-s + (0.327 − 0.188i)7-s + 0.353i·8-s + (−0.197 + 0.341i)10-s + (−0.426 − 0.246i)11-s + (0.999 − 0.0325i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (0.674 + 1.16i)17-s + (−0.972 + 0.561i)19-s + (−0.241 + 0.139i)20-s + (−0.174 − 0.301i)22-s + (−0.591 + 1.02i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0453 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0453 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0453 - 0.998i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.0453 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.461814344\)
\(L(\frac12)\) \(\approx\) \(2.461814344\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.117i)T \)
good5 \( 1 - 1.24iT - 5T^{2} \)
11 \( 1 + (1.41 + 0.816i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.78 - 4.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.23 - 2.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.83 - 4.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.59 + 2.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.53iT - 31T^{2} \)
37 \( 1 + (-6.12 - 3.53i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.46 + 1.99i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.69 - 4.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 - 4.82T + 53T^{2} \)
59 \( 1 + (-4.48 + 2.58i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.97 + 5.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.48 + 3.16i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.3 - 6.55i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.62iT - 73T^{2} \)
79 \( 1 + 8.21T + 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + (2.16 + 1.24i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.02 + 5.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.676816952126100358743083379194, −8.361856304879810240143546192545, −8.078446323739775029456106598048, −7.13249785650964184094954230664, −6.01396277057467505980769729085, −5.89230471442842773730268522794, −4.48169946574783798137636065767, −3.77353508826973603520198506479, −2.83975729029745390228057863423, −1.52465239149124540238239125374, 0.810657768757909400949990232802, 2.12734354736445638332926959735, 3.11528571997974872676595948505, 4.28531562306595367146171393524, 4.92524677592277283750552683786, 5.72380747483830125574949962649, 6.64614985799975753957355247984, 7.52714160299403163642333029078, 8.650307047228214839333451232772, 8.959534684207563036570005695618

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