Properties

Label 2-1638-39.8-c1-0-0
Degree $2$
Conductor $1638$
Sign $-0.921 - 0.388i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.904 − 0.904i)5-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + 1.27i·10-s + (1.74 + 1.74i)11-s + (−2.37 + 2.70i)13-s − 1.00i·14-s − 1.00·16-s − 5.62·17-s + (−0.234 − 0.234i)19-s + (−0.904 − 0.904i)20-s − 2.47·22-s + 2.47·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.404 − 0.404i)5-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.404i·10-s + (0.526 + 0.526i)11-s + (−0.659 + 0.751i)13-s − 0.267i·14-s − 0.250·16-s − 1.36·17-s + (−0.0537 − 0.0537i)19-s + (−0.202 − 0.202i)20-s − 0.526·22-s + 0.515·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.921 - 0.388i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.921 - 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5921116867\)
\(L(\frac12)\) \(\approx\) \(0.5921116867\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (2.37 - 2.70i)T \)
good5 \( 1 + (-0.904 + 0.904i)T - 5iT^{2} \)
11 \( 1 + (-1.74 - 1.74i)T + 11iT^{2} \)
17 \( 1 + 5.62T + 17T^{2} \)
19 \( 1 + (0.234 + 0.234i)T + 19iT^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 + (4.21 + 4.21i)T + 31iT^{2} \)
37 \( 1 + (7.99 - 7.99i)T - 37iT^{2} \)
41 \( 1 + (-3.27 + 3.27i)T - 41iT^{2} \)
43 \( 1 - 12.8iT - 43T^{2} \)
47 \( 1 + (7.44 + 7.44i)T + 47iT^{2} \)
53 \( 1 + 4.82iT - 53T^{2} \)
59 \( 1 + (-8.27 - 8.27i)T + 59iT^{2} \)
61 \( 1 - 7.19T + 61T^{2} \)
67 \( 1 + (-2.14 - 2.14i)T + 67iT^{2} \)
71 \( 1 + (4.36 - 4.36i)T - 71iT^{2} \)
73 \( 1 + (10.5 - 10.5i)T - 73iT^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + (8.36 - 8.36i)T - 83iT^{2} \)
89 \( 1 + (-10.5 - 10.5i)T + 89iT^{2} \)
97 \( 1 + (4.86 + 4.86i)T + 97iT^{2} \)
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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

−9.600614812054991457533844836400, −8.961162424431118642248454883347, −8.336054663103087826646719485188, −7.07958332344533672647467556449, −6.78530220938299192366720407155, −5.75664649437884924543845649400, −4.88979033303256939572432222983, −4.07406688907770889370841688401, −2.49356890950182872921375450198, −1.53624753720587823520180259950, 0.25768388482286359833074771889, 1.80467518076106058725306217681, 2.84166806385453866042909069327, 3.68824109197867744725535321252, 4.80586278729744763711105405534, 5.86841630764426469701890695902, 6.84261787997009321349836688905, 7.33234166020510305308314958171, 8.570551021463695158231998800047, 8.991231689641162073209156098125

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