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Properties

Label 2-1638-39.8-c1-0-14
Degree $2$
Conductor $1638$
Sign $0.314 + 0.949i$
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 + (−0.746 + 0.746i)5-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s − 1.05i·10-s + (−3.35 − 3.35i)11-s + (−1.20 + 3.39i)13-s − 1.00i·14-s − 1.00·16-s − 2.02·17-s + (5.03 + 5.03i)19-s + (0.746 + 0.746i)20-s + 4.74·22-s + 2.39·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.333 + 0.333i)5-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s − 0.333i·10-s + (−1.01 − 1.01i)11-s + (−0.335 + 0.942i)13-s − 0.267i·14-s − 0.250·16-s − 0.491·17-s + (1.15 + 1.15i)19-s + (0.166 + 0.166i)20-s + 1.01·22-s + 0.498·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.314 + 0.949i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.314 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4994731421\)
\(L(\frac12)\) \(\approx\) \(0.4994731421\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 - 3.39i)T \)
good5 \( 1 + (0.746 - 0.746i)T - 5iT^{2} \)
11 \( 1 + (3.35 + 3.35i)T + 11iT^{2} \)
17 \( 1 + 2.02T + 17T^{2} \)
19 \( 1 + (-5.03 - 5.03i)T + 19iT^{2} \)
23 \( 1 - 2.39T + 23T^{2} \)
29 \( 1 + 8.46iT - 29T^{2} \)
31 \( 1 + (5.55 + 5.55i)T + 31iT^{2} \)
37 \( 1 + (-6.97 + 6.97i)T - 37iT^{2} \)
41 \( 1 + (2.39 - 2.39i)T - 41iT^{2} \)
43 \( 1 + 9.24iT - 43T^{2} \)
47 \( 1 + (3.48 + 3.48i)T + 47iT^{2} \)
53 \( 1 + 1.35iT - 53T^{2} \)
59 \( 1 + (4.49 + 4.49i)T + 59iT^{2} \)
61 \( 1 + 1.56T + 61T^{2} \)
67 \( 1 + (1.78 + 1.78i)T + 67iT^{2} \)
71 \( 1 + (9.95 - 9.95i)T - 71iT^{2} \)
73 \( 1 + (3.90 - 3.90i)T - 73iT^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \)
89 \( 1 + (-4.40 - 4.40i)T + 89iT^{2} \)
97 \( 1 + (11.0 + 11.0i)T + 97iT^{2} \)
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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

−9.281270485937263378197917996186, −8.306573966388852181249842752571, −7.65433520679748637950189972420, −7.00637995913673126407897857136, −5.91532481473805266043549850053, −5.46972099873193593103227085901, −4.18246186828257583251616426434, −3.15343718482670346697441225573, −2.00215552361333545406039560412, −0.25212412007612913592994940116, 1.13019667006094934053782446031, 2.61703343588113328666199953456, 3.27898602585492091476206416538, 4.73394615241969652415866755421, 5.05695024363194657912707869922, 6.55966512751762063844175643822, 7.44694295372200071678934841324, 7.85946344456637384700680523094, 8.921495757349748065871049577363, 9.510681695699218194496111369647

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