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Properties

Label 2-1638-7.4-c1-0-2
Degree $2$
Conductor $1638$
Sign $-0.820 + 0.572i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.246 − 0.426i)5-s + (−0.967 + 2.46i)7-s − 0.999·8-s + (0.246 − 0.426i)10-s + (−2.23 + 3.86i)11-s − 13-s + (−2.61 + 0.392i)14-s + (−0.5 − 0.866i)16-s + (1.51 − 2.62i)17-s + (1.21 + 2.10i)19-s + 0.492·20-s − 4.46·22-s + (−0.894 − 1.54i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.110 − 0.190i)5-s + (−0.365 + 0.930i)7-s − 0.353·8-s + (0.0778 − 0.134i)10-s + (−0.672 + 1.16i)11-s − 0.277·13-s + (−0.699 + 0.105i)14-s + (−0.125 − 0.216i)16-s + (0.368 − 0.637i)17-s + (0.278 + 0.482i)19-s + 0.110·20-s − 0.951·22-s + (−0.186 − 0.323i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.820 + 0.572i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.820 + 0.572i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5781922980\)
\(L(\frac12)\) \(\approx\) \(0.5781922980\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (0.967 - 2.46i)T \)
13 \( 1 + T \)
good5 \( 1 + (0.246 + 0.426i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.23 - 3.86i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.51 + 2.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.21 - 2.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.894 + 1.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.72T + 29T^{2} \)
31 \( 1 + (3.48 - 6.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.79 + 4.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.80T + 41T^{2} \)
43 \( 1 + 4.59T + 43T^{2} \)
47 \( 1 + (5.79 + 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.06 + 12.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.13 - 7.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.11 - 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.82 - 6.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 + (8.35 - 14.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.78 - 8.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.0831T + 83T^{2} \)
89 \( 1 + (1.83 + 3.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.88T + 97T^{2} \)
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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

−9.846655049581587914228942876768, −8.862349621242581838748115110987, −8.301058981918182599818178408107, −7.24394160079459196146402309300, −6.82985536861327694492989468292, −5.47871954909763436705503711054, −5.26621922433345173529411339353, −4.14317739789373755171398301615, −3.03369617103156620247526663509, −2.02599256566952533860198506943, 0.18761037675121137336726257896, 1.58588959526327203954119827255, 3.09633663693186945821286832051, 3.51332945587502180496536959919, 4.63010743041824566302632807093, 5.56052016771768249715561307304, 6.34809537267442797173312175060, 7.39351272276145785436409404052, 8.028688496655698393897979406588, 9.123032947937848745034910566046

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