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Properties

Label 2-1638-13.10-c1-0-18
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 + (3.23 − 1.86i)11-s + (−0.866 + 3.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (−0.133 + 0.232i)17-s + (3.86 + 2.23i)19-s + (0.633 + 0.366i)20-s + (1.86 − 3.23i)22-s + (−1.73 − 3i)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 + (0.974 − 0.562i)11-s + (−0.240 + 0.970i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.0324 + 0.0562i)17-s + (0.886 + 0.512i)19-s + (0.141 + 0.0818i)20-s + (0.397 − 0.689i)22-s + (−0.361 − 0.625i)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

\(L(1)\) \(\approx\) \(2.750258754\)
\(L(\frac12)\) \(\approx\) \(2.750258754\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 + 1.86i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.133 - 0.232i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.86 - 2.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.66iT - 31T^{2} \)
37 \( 1 + (-1.09 + 0.633i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.06 + 3.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.366 + 0.633i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.46iT - 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (0.803 + 0.464i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.86 + 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.1 - 6.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.90 + 1.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.53iT - 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + 5.66iT - 83T^{2} \)
89 \( 1 + (5.59 - 3.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.633 - 0.366i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.301775824541331496088061455174, −8.764943701750530599666094575509, −7.62812638722583085043280304320, −6.75576668788393758621570743412, −6.11184067962316252122264481034, −5.15154076030173980023054868720, −4.23679128141635827233325340439, −3.43688039242239385669896405475, −2.35630445313236814519010726077, −1.21094993674578404159716550300, 1.09635316575717931541341204383, 2.51548100720292289646161254086, 3.63783633997904440292916259931, 4.48742666992015819705964100624, 5.28515527781232310540471930832, 6.04240685948701470741419830826, 7.13794360000943123884101854617, 7.57119862756372242509886631496, 8.552191533719534780887513450056, 9.357055562636567038969532098530

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