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Properties

Label 2-1638-91.81-c1-0-0
Degree $2$
Conductor $1638$
Sign $0.411 - 0.911i$
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 + (−1.69 − 2.93i)5-s + (−2.60 − 0.461i)7-s − 0.999·8-s − 3.38·10-s − 3.81·11-s + (−3.24 − 1.58i)13-s + (−1.70 + 2.02i)14-s + (−0.5 + 0.866i)16-s + (2.02 + 3.50i)17-s + 6.15·19-s + (−1.69 + 2.93i)20-s + (−1.90 + 3.30i)22-s + (0.528 − 0.915i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.757 − 1.31i)5-s + (−0.984 − 0.174i)7-s − 0.353·8-s − 1.07·10-s − 1.14·11-s + (−0.898 − 0.438i)13-s + (−0.454 + 0.541i)14-s + (−0.125 + 0.216i)16-s + (0.490 + 0.849i)17-s + 1.41·19-s + (−0.378 + 0.655i)20-s + (−0.406 + 0.703i)22-s + (0.110 − 0.190i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.411 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07538619574\)
\(L(\frac12)\) \(\approx\) \(0.07538619574\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 + 0.461i)T \)
13 \( 1 + (3.24 + 1.58i)T \)
good5 \( 1 + (1.69 + 2.93i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.81T + 11T^{2} \)
17 \( 1 + (-2.02 - 3.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.15T + 19T^{2} \)
23 \( 1 + (-0.528 + 0.915i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.15 - 1.99i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.99 - 3.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.71 + 4.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.26 - 3.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.46 + 6.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.77 - 4.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.81 - 6.61i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.74 + 6.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 + 8.84T + 67T^{2} \)
71 \( 1 + (-7.21 + 12.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.61 - 6.26i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.222 - 0.386i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 + (-4.11 + 7.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.18 - 10.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.533717727462940211315303777960, −8.930922220788505329437003349873, −7.85112983692257124016675572105, −7.43063818888463540086021441548, −5.97187040467715723321960266872, −5.23355398279347274021570323169, −4.54774352195865548075666445533, −3.52543848408957372461770919550, −2.74258093063510091048055894049, −1.12290310702531925747918653060, 0.02871019862779360191194071411, 2.81185570492169464169561770092, 2.99008999448205740068263827874, 4.17630290079328279353637342721, 5.25827389726553051650598037930, 6.04389234609959467017324478814, 7.11913419336881634159395484042, 7.32692880175807220354817718222, 8.061474892768406310456614891963, 9.413217683719848844008175348690

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