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Properties

Label 2-1638-13.3-c1-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 4·5-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (2 + 3.46i)10-s + (−1.5 − 2.59i)11-s + (−2.5 − 2.59i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (−2.5 + 4.33i)17-s + (−1.5 + 2.59i)19-s + (1.99 − 3.46i)20-s + (−1.5 + 2.59i)22-s + (3 + 5.19i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.78·5-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (0.632 + 1.09i)10-s + (−0.452 − 0.783i)11-s + (−0.693 − 0.720i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + (−0.344 + 0.596i)19-s + (0.447 − 0.774i)20-s + (−0.319 + 0.553i)22-s + (0.625 + 1.08i)23-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: $\chi_{1638} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5432283619\)
\(L(\frac12)\) \(\approx\) \(0.5432283619\)
\(L(\frac{3}{2})\) 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 \( 1 + 4T + 5T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 - 11T + 53T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12T + 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6 + 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.209572250965169719824095895726, −8.455029091833649262596810241522, −7.83850930514762571768700933667, −7.35076259395235782952250669161, −6.04490230441056696329922628557, −5.02721514047159016780863977095, −3.87917850940955487133888764526, −3.47043946501003338384843077762, −2.29583356565372362450505337540, −0.53510965186498306838775895806, 0.50096710904694089067719525983, 2.43221517391596455348626634829, 3.71265008499387185484580174349, 4.61205922556751389052858201972, 5.06776879853658935663386643013, 6.76724997137839378946880176352, 7.16388363282129972261873711635, 7.58075107221998171421476051498, 8.730989720272748181507728750806, 9.076285994286354126864852649807

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