↑

Properties

Label 2-1638-39.5-c1-0-9
Degree $2$
Conductor $1638$
Sign $-0.934 - 0.354i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (1.64 + 1.64i)5-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + 2.32i·10-s + (−1.53 + 1.53i)11-s + (−0.148 + 3.60i)13-s − 1.00i·14-s − 1.00·16-s − 5.58·17-s + (−0.269 + 0.269i)19-s + (−1.64 + 1.64i)20-s − 2.16·22-s − 4.44·23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.734 + 0.734i)5-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + 0.734i·10-s + (−0.461 + 0.461i)11-s + (−0.0412 + 0.999i)13-s − 0.267i·14-s − 0.250·16-s − 1.35·17-s + (−0.0618 + 0.0618i)19-s + (−0.367 + 0.367i)20-s − 0.461·22-s − 0.927·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.934 - 0.354i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.934 - 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.683022144\)
\(L(\frac12)\) \(\approx\) \(1.683022144\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.148 - 3.60i)T \)
good5 \( 1 + (-1.64 - 1.64i)T + 5iT^{2} \)
11 \( 1 + (1.53 - 1.53i)T - 11iT^{2} \)
17 \( 1 + 5.58T + 17T^{2} \)
19 \( 1 + (0.269 - 0.269i)T - 19iT^{2} \)
23 \( 1 + 4.44T + 23T^{2} \)
29 \( 1 - 6.10iT - 29T^{2} \)
31 \( 1 + (0.476 - 0.476i)T - 31iT^{2} \)
37 \( 1 + (-2.38 - 2.38i)T + 37iT^{2} \)
41 \( 1 + (-7.88 - 7.88i)T + 41iT^{2} \)
43 \( 1 + 4.01iT - 43T^{2} \)
47 \( 1 + (-1.17 + 1.17i)T - 47iT^{2} \)
53 \( 1 + 2.66iT - 53T^{2} \)
59 \( 1 + (1.84 - 1.84i)T - 59iT^{2} \)
61 \( 1 + 1.03T + 61T^{2} \)
67 \( 1 + (2.08 - 2.08i)T - 67iT^{2} \)
71 \( 1 + (-5.32 - 5.32i)T + 71iT^{2} \)
73 \( 1 + (5.80 + 5.80i)T + 73iT^{2} \)
79 \( 1 + 2.05T + 79T^{2} \)
83 \( 1 + (-8.82 - 8.82i)T + 83iT^{2} \)
89 \( 1 + (4.84 - 4.84i)T - 89iT^{2} \)
97 \( 1 + (-0.753 + 0.753i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.687727664636753655772576192563, −8.976026323831545347702933980582, −7.984879546527643611194949580305, −7.01001924348101875310651324229, −6.58770713163357250277629173016, −5.84633225465137455763854805621, −4.74515773677804379267293128401, −4.02451989060877895949607656181, −2.78529261978196394161574677150, −1.96368000810235260659933098462, 0.49781743218673871674366705945, 1.99273755276094991976129230547, 2.78521616249389930839393438223, 4.00282825074444457381294109183, 4.88346119440865570725705405193, 5.80137380584502177014719168726, 6.12858252887879550547386658072, 7.48050757807446341493658942063, 8.415040132856186788888752768284, 9.177890381517081584610090000660

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