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Properties

Label 2-1638-91.74-c1-0-6
Degree $2$
Conductor $1638$
Sign $0.200 - 0.979i$
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  − 2-s + 4-s + (−0.611 − 1.05i)5-s + (1.15 + 2.38i)7-s − 8-s + (0.611 + 1.05i)10-s + (0.0702 + 0.121i)11-s + (2.39 + 2.69i)13-s + (−1.15 − 2.38i)14-s + 16-s − 0.186·17-s + (−0.447 + 0.775i)19-s + (−0.611 − 1.05i)20-s + (−0.0702 − 0.121i)22-s − 0.0364·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.273 − 0.473i)5-s + (0.435 + 0.900i)7-s − 0.353·8-s + (0.193 + 0.334i)10-s + (0.0211 + 0.0367i)11-s + (0.663 + 0.747i)13-s + (−0.307 − 0.636i)14-s + 0.250·16-s − 0.0452·17-s + (−0.102 + 0.177i)19-s + (−0.136 − 0.236i)20-s + (−0.0149 − 0.0259i)22-s − 0.00760·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.200 - 0.979i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.200 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.017514650\)
\(L(\frac12)\) \(\approx\) \(1.017514650\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 + (-1.15 - 2.38i)T \)
13 \( 1 + (-2.39 - 2.69i)T \)
good5 \( 1 + (0.611 + 1.05i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.0702 - 0.121i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.186T + 17T^{2} \)
19 \( 1 + (0.447 - 0.775i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.0364T + 23T^{2} \)
29 \( 1 + (2.99 - 5.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.82 - 3.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.363T + 37T^{2} \)
41 \( 1 + (1.70 - 2.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.06 + 3.58i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.358 + 0.621i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.49 - 6.05i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.99T + 59T^{2} \)
61 \( 1 + (0.186 - 0.323i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.42 - 4.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.31 - 9.20i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.80 - 8.32i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.94 + 5.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.14T + 83T^{2} \)
89 \( 1 - 6.35T + 89T^{2} \)
97 \( 1 + (-3.24 - 5.61i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.298652172722706255871290803411, −8.702724648161327591531761937232, −8.297151514639235178830203500711, −7.26710162673752725235726461038, −6.43484183310565275610834775418, −5.53405085135009188198361323884, −4.65322868323322810688727025342, −3.52869125071885916129857165286, −2.27479833596973236567363814654, −1.27649455352721273658798512533, 0.53723574596187774795764734625, 1.83454194351180220698543867826, 3.17541375850223219713077985092, 3.96147254319672791967769607895, 5.14653985917108461049105395369, 6.19072678225689991466933567354, 6.99951222065683462507606843393, 7.74569378994468396470205155070, 8.242257656090244168147278532406, 9.243081740120116113247634804020

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