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Properties

Label 2-1638-13.9-c1-0-24
Degree $2$
Conductor $1638$
Sign $-0.859 + 0.511i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 3·5-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (1.5 − 2.59i)10-s + (2.5 − 2.59i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + (2 + 3.46i)19-s + (1.49 + 2.59i)20-s + (1.5 − 2.59i)23-s + 4·25-s + (1 + 3.46i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.34·5-s + (−0.188 − 0.327i)7-s + 0.353·8-s + (0.474 − 0.821i)10-s + (0.693 − 0.720i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + (0.458 + 0.794i)19-s + (0.335 + 0.580i)20-s + (0.312 − 0.541i)23-s + 0.800·25-s + (0.196 + 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.859 + 0.511i$
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: \(1\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 2.59i)T \)
good5 \( 1 + 3T + 5T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.5 + 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.748276987373958787166041643777, −8.022530740941099095309154796721, −7.76925061340699859271400623055, −6.73095173934942509089957992803, −5.94706332934839104246854760940, −4.97777727641876856286263336221, −3.84423679172735358704651473898, −3.37587448310447171637300057402, −1.40320390684268550963506944719, 0, 1.43660887997356190527905534323, 2.99602865912465226327249212230, 3.54733071740672976248642527689, 4.57391976956826796824376253699, 5.43680035632952664373421911823, 6.85286070345043910990162241056, 7.40330414381508375584473572646, 8.195365691898283356273906745321, 9.115774427588332292137399176638

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