Properties

Label 2-1638-273.68-c1-0-1
Degree $2$
Conductor $1638$
Sign $-0.0138 - 0.999i$
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  i·2-s − 4-s + (0.485 − 0.841i)5-s + (−2.47 − 0.940i)7-s + i·8-s + (−0.841 − 0.485i)10-s + (−2.28 − 1.32i)11-s + (−2.23 − 2.83i)13-s + (−0.940 + 2.47i)14-s + 16-s + 0.627·17-s + (0.415 − 0.239i)19-s + (−0.485 + 0.841i)20-s + (−1.32 + 2.28i)22-s + 3.48i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.217 − 0.376i)5-s + (−0.934 − 0.355i)7-s + 0.353i·8-s + (−0.266 − 0.153i)10-s + (−0.689 − 0.398i)11-s + (−0.619 − 0.785i)13-s + (−0.251 + 0.660i)14-s + 0.250·16-s + 0.152·17-s + (0.0952 − 0.0550i)19-s + (−0.108 + 0.188i)20-s + (−0.281 + 0.487i)22-s + 0.727i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0138 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0138 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0138 - 0.999i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.0138 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1137701242\)
\(L(\frac12)\) \(\approx\) \(0.1137701242\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (2.47 + 0.940i)T \)
13 \( 1 + (2.23 + 2.83i)T \)
good5 \( 1 + (-0.485 + 0.841i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.28 + 1.32i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.627T + 17T^{2} \)
19 \( 1 + (-0.415 + 0.239i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.48iT - 23T^{2} \)
29 \( 1 + (4.77 - 2.75i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.59 + 0.920i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.51T + 37T^{2} \)
41 \( 1 + (-0.880 - 1.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.424 - 0.734i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.32 - 7.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.50 - 4.90i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.11T + 59T^{2} \)
61 \( 1 + (11.0 - 6.37i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.40 - 4.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.14 + 5.28i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-11.8 + 6.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.04 + 1.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 - 8.12T + 89T^{2} \)
97 \( 1 + (-2.72 - 1.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.479598618873572609932849527309, −9.209362843848786165315753459809, −7.948937189434197772476188291688, −7.42193035481392821087477315953, −6.16403449256932172564167078426, −5.42101012711871152667957787594, −4.56023011846112468172838239124, −3.34435545825892196266987904628, −2.80851953876540858286444290535, −1.31253405325242242582878038763, 0.04418063177742970791487921254, 2.16374997367262269874734215794, 3.11734990449771286917717456367, 4.31056270645676956808305691230, 5.16635225469864645887480750170, 6.14000998262612866292779901816, 6.69853002300918193369201935456, 7.45852959123407108552463506532, 8.316575140279138555849677932189, 9.235592393300205119039148887825

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