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Properties

Label 2-1638-13.10-c1-0-10
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: $\chi_{1638} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.0453 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.274991996\)
\(L(\frac12)\) \(\approx\) \(1.274991996\)
\(L(\frac{3}{2})\) 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

−9.265884128020271733089793653463, −8.850763816037150720322655432765, −8.062817365562773409663291278456, −7.20492852639242843313431630023, −6.33593770497694945790199336616, −5.89729264028420717674836552248, −4.60386170775407703592111064343, −3.63024019558302703076740925005, −2.39903115219632132371289198084, −1.23540456400531634060342546835, 0.68648295231166232757845933070, 1.80557920110899329390622678186, 2.98361541753427380116195455624, 4.20082061459965237542413678210, 4.82329642838064957510253385212, 6.13619842343519846845708320135, 6.84283370390457624343750276663, 7.80653791941733618847869241175, 8.616399129344277415818045898019, 9.046534446176418942107006654434

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