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Properties

Label 2-1638-13.10-c1-0-25
Degree $2$
Conductor $1638$
Sign $0.287 + 0.957i$
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 − 1.25i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (0.627 + 1.08i)10-s + (4.35 − 2.51i)11-s + (−3.43 + 1.08i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (2.26 − 3.91i)17-s + (−3.05 − 1.76i)19-s + (−1.08 − 0.627i)20-s + (−2.51 + 4.35i)22-s + (−2.56 − 4.44i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.561i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (0.198 + 0.343i)10-s + (1.31 − 0.757i)11-s + (−0.953 + 0.299i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.548 − 0.949i)17-s + (−0.700 − 0.404i)19-s + (−0.243 − 0.140i)20-s + (−0.535 + 0.928i)22-s + (−0.535 − 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.287 + 0.957i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.287 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.124423492\)
\(L(\frac12)\) \(\approx\) \(1.124423492\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.43 - 1.08i)T \)
good5 \( 1 + 1.25iT - 5T^{2} \)
11 \( 1 + (-4.35 + 2.51i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.26 + 3.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.05 + 1.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.56 + 4.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.765 - 1.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.439iT - 31T^{2} \)
37 \( 1 + (8.90 - 5.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.65 + 4.99i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.83 - 3.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.60iT - 47T^{2} \)
53 \( 1 + 8.82T + 53T^{2} \)
59 \( 1 + (-0.542 - 0.313i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.49 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.40 + 3.69i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.89 + 3.40i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.5iT - 73T^{2} \)
79 \( 1 + 6.74T + 79T^{2} \)
83 \( 1 + 9.57iT - 83T^{2} \)
89 \( 1 + (-7.28 + 4.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.43 + 1.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.042303568097802109442459921568, −8.624307915318552219475469459179, −7.71939991882906642577516743846, −6.82404768351502607700422487191, −6.17809075574567712304197797033, −5.07913960842831024193309362748, −4.43998190289900610182154746102, −3.05834199374618760865495683949, −1.78751009987588553996505260975, −0.56020374019418857421822118974, 1.37566802431133529661062055954, 2.31384888307251556914184640873, 3.57724943123462653509897394998, 4.28305775953619292799277699795, 5.53979410659511598123027094602, 6.59678534884718205820060887593, 7.21642504759040086178011827870, 7.977081832223158844724820017107, 8.815250495253581740176245987184, 9.699613058796789436255358358449

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