↑

Properties

Label 2-1682-29.28-c1-0-31
Degree $2$
Conductor $1682$
Sign $-0.989 - 0.141i$
Analytic cond. $13.4308$
Root an. cond. $3.66481$
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  + i·2-s + 2.88i·3-s − 4-s + 2.81·5-s − 2.88·6-s + 3.85·7-s − i·8-s − 5.35·9-s + 2.81i·10-s + 2.90i·11-s − 2.88i·12-s + 0.364·13-s + 3.85i·14-s + 8.12i·15-s + 16-s − 0.925i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.66i·3-s − 0.5·4-s + 1.25·5-s − 1.17·6-s + 1.45·7-s − 0.353i·8-s − 1.78·9-s + 0.889i·10-s + 0.876i·11-s − 0.834i·12-s + 0.101·13-s + 1.02i·14-s + 2.09i·15-s + 0.250·16-s − 0.224i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1682\)    =    \(2 \cdot 29^{2}\)
Sign: $-0.989 - 0.141i$
Analytic conductor: \(13.4308\)
Root analytic conductor: \(3.66481\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1682} (1681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1682,\ (\ :1/2),\ -0.989 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.272996751\)
\(L(\frac12)\) \(\approx\) \(2.272996751\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
29 \( 1 \)
good3 \( 1 - 2.88iT - 3T^{2} \)
5 \( 1 - 2.81T + 5T^{2} \)
7 \( 1 - 3.85T + 7T^{2} \)
11 \( 1 - 2.90iT - 11T^{2} \)
13 \( 1 - 0.364T + 13T^{2} \)
17 \( 1 + 0.925iT - 17T^{2} \)
19 \( 1 + 5.15iT - 19T^{2} \)
23 \( 1 + 2.53T + 23T^{2} \)
31 \( 1 - 8.03iT - 31T^{2} \)
37 \( 1 - 3.62iT - 37T^{2} \)
41 \( 1 + 4.99iT - 41T^{2} \)
43 \( 1 - 8.55iT - 43T^{2} \)
47 \( 1 - 7.50iT - 47T^{2} \)
53 \( 1 + 9.16T + 53T^{2} \)
59 \( 1 + 4.66T + 59T^{2} \)
61 \( 1 - 6.85iT - 61T^{2} \)
67 \( 1 - 1.40T + 67T^{2} \)
71 \( 1 - 4.31T + 71T^{2} \)
73 \( 1 - 12.5iT - 73T^{2} \)
79 \( 1 + 12.0iT - 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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

−9.617397322650222683943807783940, −9.082776953316463275870123612334, −8.372636027775550897483432205449, −7.38441643697355189228920853813, −6.29533982985320044601714023126, −5.40906125250230330894555940311, −4.77303665223866491057415088614, −4.44582938694923730169300578471, −2.98803071366726838731385772621, −1.70508040777107485058005780019, 0.898344286791321625907857709007, 1.92812615418588373601155604097, 2.13709702147950374047655033403, 3.63360343215276494722032065958, 5.07127992496119563403588717908, 5.86924895764484201513929794542, 6.33649293421323134509840427149, 7.69322815596617329915033735389, 8.073161155591694097760190676853, 8.827925399722118377653916982609

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