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Properties

Label 2-1638-13.9-c1-0-6
Degree $2$
Conductor $1638$
Sign $-0.872 - 0.488i$
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 + 2·5-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−1 + 1.73i)10-s + (−2 + 3.46i)11-s + (−3.5 − 0.866i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (3.5 + 6.06i)17-s + (−1 − 1.73i)19-s + (−0.999 − 1.73i)20-s + (−1.99 − 3.46i)22-s + (−0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.894·5-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.316 + 0.547i)10-s + (−0.603 + 1.04i)11-s + (−0.970 − 0.240i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (0.848 + 1.47i)17-s + (−0.229 − 0.397i)19-s + (−0.223 − 0.387i)20-s + (−0.426 − 0.738i)22-s + (−0.104 + 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.872 - 0.488i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.045116497\)
\(L(\frac12)\) \(\approx\) \(1.045116497\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (3.5 + 0.866i)T \)
good5 \( 1 - 2T + 5T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9T + 31T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-7.5 - 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.669533033871346904008043128175, −9.006570598722941376458067428915, −7.966622955765090332313248848926, −7.45397594130239479069616350947, −6.50666402101929150346338032102, −5.57367479694585242909144318197, −5.17096030392605402313517005075, −3.98652754648966586795570880068, −2.46011244694000534717974466018, −1.63513405545175008033898498993, 0.42698813041559905090219435230, 1.86448015206884076550505144257, 2.76527997354497507826947511820, 3.73934886432153024356773800070, 5.05077766017886537264560511044, 5.56227794120664417953217923564, 6.73305917898861959478450693490, 7.64279738716173650968807363233, 8.276296823344769309324498057909, 9.460471315512146920822304815418

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