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Properties

Label 2-1638-13.10-c1-0-23
Degree $2$
Conductor $1638$
Sign $0.419 + 0.907i$
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 − 3.05i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (1.52 + 2.64i)10-s + (2.98 − 1.72i)11-s + (3.25 − 1.55i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (1.41 − 2.44i)17-s + (1.49 + 0.862i)19-s + (−2.64 − 1.52i)20-s + (−1.72 + 2.98i)22-s + (−1.53 − 2.65i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 1.36i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (0.483 + 0.836i)10-s + (0.898 − 0.518i)11-s + (0.902 − 0.431i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.343 − 0.594i)17-s + (0.342 + 0.197i)19-s + (−0.591 − 0.341i)20-s + (−0.366 + 0.635i)22-s + (−0.319 − 0.553i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.419 + 0.907i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.419 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.425845836\)
\(L(\frac12)\) \(\approx\) \(1.425845836\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.25 + 1.55i)T \)
good5 \( 1 + 3.05iT - 5T^{2} \)
11 \( 1 + (-2.98 + 1.72i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.41 + 2.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.49 - 0.862i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.53 + 2.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.92 - 3.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.978iT - 31T^{2} \)
37 \( 1 + (-4.58 + 2.64i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.62 - 4.98i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.51 - 2.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.04iT - 47T^{2} \)
53 \( 1 - 8.33T + 53T^{2} \)
59 \( 1 + (8.64 + 4.98i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.77 + 9.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.11 + 2.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.17 - 1.83i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 4.49T + 79T^{2} \)
83 \( 1 - 7.15iT - 83T^{2} \)
89 \( 1 + (-6.84 + 3.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.88 + 4.54i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.169371347876956374465194817525, −8.319904115835358225297619111674, −8.108657602204317950577141389559, −6.83460779957128155255915626031, −5.98871710116246763032637161638, −5.23708621441581528634199363297, −4.40473692229841694767368298466, −3.21325326088267648399852447054, −1.56985367691734936039870427237, −0.78667043121624930673587442428, 1.34799780408921185919779561263, 2.38585235550915691570208566319, 3.54134925062431481919316179599, 4.11920407765746561208177629165, 5.65450961735001072895215586486, 6.65073504214617368971132776691, 7.04081775513868856248831005029, 7.988765963411748305367563121158, 8.760408771434545661768903707342, 9.683748998090173868128419899773

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