↑

Properties

Label 2-182-7.2-c1-0-1
Degree $2$
Conductor $182$
Sign $-0.569 - 0.822i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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  + (−0.5 + 0.866i)2-s + (1.46 + 2.54i)3-s + (−0.499 − 0.866i)4-s + (−0.846 + 1.46i)5-s − 2.93·6-s + (2.26 − 1.36i)7-s + 0.999·8-s + (−2.81 + 4.87i)9-s + (−0.846 − 1.46i)10-s + (−0.202 − 0.350i)11-s + (1.46 − 2.54i)12-s − 13-s + (0.0490 + 2.64i)14-s − 4.97·15-s + (−0.5 + 0.866i)16-s + (−2.51 − 4.36i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.848 + 1.46i)3-s + (−0.249 − 0.433i)4-s + (−0.378 + 0.655i)5-s − 1.19·6-s + (0.856 − 0.515i)7-s + 0.353·8-s + (−0.938 + 1.62i)9-s + (−0.267 − 0.463i)10-s + (−0.0610 − 0.105i)11-s + (0.424 − 0.734i)12-s − 0.277·13-s + (0.0130 + 0.706i)14-s − 1.28·15-s + (−0.125 + 0.216i)16-s + (−0.610 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $-0.569 - 0.822i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ -0.569 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.563257 + 1.07495i\)
\(L(\frac12)\) \(\approx\) \(0.563257 + 1.07495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.26 + 1.36i)T \)
13 \( 1 + T \)
good3 \( 1 + (-1.46 - 2.54i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.846 - 1.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.202 + 0.350i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.51 + 4.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.83 + 6.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.93 - 6.82i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.40T + 29T^{2} \)
31 \( 1 + (-0.395 - 0.685i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.46 + 2.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 2.18T + 43T^{2} \)
47 \( 1 + (1.51 - 2.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.895 + 1.55i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.14 + 8.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.04 - 1.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.202 + 0.350i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.22T + 71T^{2} \)
73 \( 1 + (7.47 + 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.47 - 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (6.47 - 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.38T + 97T^{2} \)
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

−13.65545542951823005083290292405, −11.41012008103601576567412689146, −10.90080807078055992399588971789, −9.740547793956304123285797734732, −9.104075215885544967900806923922, −7.914576521389667861058111490124, −7.14304366865934241766770964897, −5.19386932807938819683720050854, −4.31576526985185508451129662114, −2.93003229393204353666731151317, 1.38822366255646330163222338715, 2.55000212586563228838468336850, 4.31718015052860591132052674945, 6.12895217657612581184455808495, 7.65160215864409833164517871731, 8.263881755881463606291112424182, 8.834798972280861328642000352503, 10.31151524390835991305145599988, 11.78218508205610127927374656716, 12.31529873995115182789023417828

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