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Properties

Label 2-1638-7.4-c1-0-38
Degree $2$
Conductor $1638$
Sign $-0.605 - 0.795i$
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 + (−1 − 1.73i)5-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (−0.999 + 1.73i)10-s + (1.5 − 2.59i)11-s − 13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (2.5 − 4.33i)17-s + (−0.5 − 0.866i)19-s + 1.99·20-s − 3·22-s + (0.500 − 0.866i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.447 − 0.774i)5-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (−0.316 + 0.547i)10-s + (0.452 − 0.783i)11-s − 0.277·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.606 − 1.05i)17-s + (−0.114 − 0.198i)19-s + 0.447·20-s − 0.639·22-s + (0.100 − 0.173i)25-s + (0.0980 + 0.169i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.605 - 0.795i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3519825429\)
\(L(\frac12)\) \(\approx\) \(0.3519825429\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
13 \( 1 + T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - T + 71T^{2} \)
73 \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−8.807796729808393443257251890951, −8.422914580404262706192026121184, −7.31988821201935264494538366897, −6.62591983017440734858880239501, −5.43841329984929617584811183333, −4.54349973780420823816767985945, −3.55717248041186480767429172934, −2.86002270774807233345622454742, −1.22598226868610572368503868820, −0.16674946465191759335621931487, 1.79578229148379102515830178648, 3.16616201891322676865941832381, 3.94444374604613115678851810630, 5.13708092626654107699565717982, 6.14422389392412965132372362609, 6.73559753646165973976409876730, 7.39737822914420692905312622843, 8.228579190416011302502578097491, 9.081777044823750947308039423967, 9.938101877606767922038269098005

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