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Properties

Label 2-1638-7.2-c1-0-18
Degree $2$
Conductor $1638$
Sign $0.997 + 0.0762i$
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 + (0.397 − 0.687i)5-s + (2.08 + 1.62i)7-s − 0.999·8-s + (−0.397 − 0.687i)10-s + (1.92 + 3.33i)11-s − 13-s + (2.45 − 0.990i)14-s + (−0.5 + 0.866i)16-s + (1.05 + 1.82i)17-s + (−2.48 + 4.29i)19-s − 0.794·20-s + 3.85·22-s + (1.76 − 3.05i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.177 − 0.307i)5-s + (0.787 + 0.615i)7-s − 0.353·8-s + (−0.125 − 0.217i)10-s + (0.580 + 1.00i)11-s − 0.277·13-s + (0.655 − 0.264i)14-s + (−0.125 + 0.216i)16-s + (0.255 + 0.443i)17-s + (−0.569 + 0.986i)19-s − 0.177·20-s + 0.821·22-s + (0.368 − 0.637i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.997 + 0.0762i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.997 + 0.0762i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.249506222\)
\(L(\frac12)\) \(\approx\) \(2.249506222\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.08 - 1.62i)T \)
13 \( 1 + T \)
good5 \( 1 + (-0.397 + 0.687i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.92 - 3.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.05 - 1.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.48 - 4.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.76 + 3.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.70T + 29T^{2} \)
31 \( 1 + (-0.0295 - 0.0511i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.74 - 8.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.05T + 41T^{2} \)
43 \( 1 - 3.47T + 43T^{2} \)
47 \( 1 + (-0.472 + 0.817i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.92 - 3.32i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.58 + 7.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.75 + 4.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.59 + 11.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.96T + 71T^{2} \)
73 \( 1 + (-2.61 - 4.53i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.76 - 4.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + (-6.93 + 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.74T + 97T^{2} \)
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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

−9.439058342270565966970229366365, −8.671111330835199835265751900280, −7.985753705114123171765661374632, −6.84173791776394234707216011537, −5.99372865618885819011073379960, −4.96853855715166545448760078392, −4.53135557313379300718421403993, −3.35769889288227427287246912825, −2.13508708316369632451588215008, −1.38726309872589944727385064123, 0.855901755160951682018289170580, 2.49283169931232728615742120190, 3.58104032375221885346798037706, 4.50638811321496383738580915190, 5.27401810318042393512145235404, 6.23207923457982824921025977650, 6.98100130607546888150237262292, 7.63066550327187646451736835148, 8.600636881043558441472157307834, 9.083684303598466979787685247337

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