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Properties

Label 2-1638-13.10-c1-0-11
Degree $2$
Conductor $1638$
Sign $-0.183 - 0.983i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 2.19i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−1.09 − 1.89i)10-s + (−3.54 + 2.04i)11-s + (3.53 − 0.706i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (2.85 − 4.93i)17-s + (6.69 + 3.86i)19-s + (1.89 + 1.09i)20-s + (2.04 − 3.54i)22-s + (−1.23 − 2.14i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.980i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.346 − 0.600i)10-s + (−1.06 + 0.616i)11-s + (0.980 − 0.196i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.691 − 1.19i)17-s + (1.53 + 0.886i)19-s + (0.424 + 0.245i)20-s + (0.435 − 0.755i)22-s + (−0.258 − 0.447i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.183 - 0.983i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.183 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.262193076\)
\(L(\frac12)\) \(\approx\) \(1.262193076\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-3.53 + 0.706i)T \)
good5 \( 1 - 2.19iT - 5T^{2} \)
11 \( 1 + (3.54 - 2.04i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.85 + 4.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.69 - 3.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.23 + 2.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.90 - 8.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.19iT - 31T^{2} \)
37 \( 1 + (8.79 - 5.07i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.14 - 1.81i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.37 - 2.38i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.70iT - 47T^{2} \)
53 \( 1 - 6.31T + 53T^{2} \)
59 \( 1 + (-4.20 - 2.42i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.63 + 9.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.49 - 3.17i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.21 + 4.16i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.37iT - 73T^{2} \)
79 \( 1 - 3.45T + 79T^{2} \)
83 \( 1 + 0.332iT - 83T^{2} \)
89 \( 1 + (11.7 - 6.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.0 - 5.80i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.781317412627081726174908603863, −8.689881531342553825310705619965, −7.904955142515421507871672871517, −7.33254350804985506683332565134, −6.59352887271357040048225028198, −5.56793445149107358329302173078, −4.95228614775219841416014050306, −3.37852648310366344705700059731, −2.64710374567470565514554430844, −1.24957970007362354644070963289, 0.69046869594014072079153430203, 1.65395991825429234863099151121, 3.06286968413610492029204148717, 3.95519820430291287159696866576, 5.15355392670660067505202292672, 5.71812305370841430985424856668, 6.97541309101079898640251358717, 7.85562987640495348077697240584, 8.576380930255167629182561114303, 8.865737608389913135183311639667

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