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Properties

Label 2-1638-13.10-c1-0-15
Degree $2$
Conductor $1638$
Sign $0.967 + 0.252i$
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 − 0.732i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (0.366 + 0.633i)10-s + (−4.5 + 2.59i)11-s + (0.866 − 3.5i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (1.13 − 1.96i)17-s + (−0.401 − 0.232i)19-s + (−0.633 − 0.366i)20-s + (2.59 − 4.5i)22-s + (3.73 + 6.46i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.327i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (0.115 + 0.200i)10-s + (−1.35 + 0.783i)11-s + (0.240 − 0.970i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.275 − 0.476i)17-s + (−0.0922 − 0.0532i)19-s + (−0.141 − 0.0818i)20-s + (0.553 − 0.959i)22-s + (0.778 + 1.34i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.967 + 0.252i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.163211644\)
\(L(\frac12)\) \(\approx\) \(1.163211644\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 3.5i)T \)
good5 \( 1 + 0.732iT - 5T^{2} \)
11 \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.13 + 1.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.401 + 0.232i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.73 - 6.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.76 + 3.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.26iT - 31T^{2} \)
37 \( 1 + (-5.83 + 3.36i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.33 + 4.23i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.36 - 7.56i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.92iT - 47T^{2} \)
53 \( 1 - 9.92T + 53T^{2} \)
59 \( 1 + (7.73 + 4.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.86 - 3.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.29 + 3.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.46iT - 73T^{2} \)
79 \( 1 - 9T + 79T^{2} \)
83 \( 1 + 9.26iT - 83T^{2} \)
89 \( 1 + (-14.5 + 8.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.2 + 7.09i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.351550597979900169783779023046, −8.492683364541132851426200725972, −7.59847621732030398201762062895, −7.43334647281825896936716202728, −6.02106453341447072626562259873, −5.32613991654985365664497739295, −4.65619463482084380411229580780, −3.14427427841601569963995780953, −2.12903865573464311449534395692, −0.70681676093335654062076323899, 0.975709295342223765033876275038, 2.36396898067471465110391944411, 3.16404193596859141302802824535, 4.32196959623544973508580373371, 5.29234027864566441476587721927, 6.38418109883951412317221690378, 7.10550817602247635044449926623, 8.041971325360596332343202525212, 8.585382099308550828999362735459, 9.346250681074585387352784394507

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