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Properties

Label 2-1638-13.9-c1-0-20
Degree $2$
Conductor $1638$
Sign $0.999 + 0.0256i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 2·5-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−1 + 1.73i)10-s + (1.5 − 2.59i)11-s + (3.5 − 0.866i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (−3.5 − 6.06i)17-s + (2.5 + 4.33i)19-s + (−0.999 − 1.73i)20-s + (1.5 + 2.59i)22-s + (3 − 5.19i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.894·5-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.316 + 0.547i)10-s + (0.452 − 0.783i)11-s + (0.970 − 0.240i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (−0.848 − 1.47i)17-s + (0.573 + 0.993i)19-s + (−0.223 − 0.387i)20-s + (0.319 + 0.553i)22-s + (0.625 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.999 + 0.0256i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.999 + 0.0256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.743090321\)
\(L(\frac12)\) \(\approx\) \(1.743090321\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 - 2T + 5T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4 - 6.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12T + 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (5.5 - 9.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.319924569728625790133066257743, −8.593056959325535491532063839661, −7.943635454885188618454364943987, −6.79801724096431136240805065578, −6.18553817099079641740629280362, −5.53067649590933348750911956363, −4.62974658536746250321058433717, −3.34854637759005914573198834269, −2.14097022106558314804100139927, −0.851929117004189493613706571754, 1.32861683691533495462521530019, 1.99984750906877831918923499742, 3.33339877802460136353048902407, 4.23377388606589454098852972375, 5.18924713622841847959653280589, 6.25925316661206356313102409951, 6.96212929606193242358741988711, 7.923180892627068648403769614174, 8.996581583738219938353598192166, 9.236037074383809865154478888425

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