Properties

Label 2-162-27.4-c3-0-3
Degree $2$
Conductor $162$
Sign $-0.130 - 0.991i$
Analytic cond. $9.55830$
Root an. cond. $3.09165$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.87 + 0.684i)2-s + (3.06 + 2.57i)4-s + (−1.31 + 7.46i)5-s + (−2.90 + 2.43i)7-s + (4.00 + 6.92i)8-s + (−7.57 + 13.1i)10-s + (8.73 + 49.5i)11-s + (−7.25 + 2.64i)13-s + (−7.11 + 2.59i)14-s + (2.77 + 15.7i)16-s + (−17.4 + 30.2i)17-s + (36.2 + 62.8i)19-s + (−23.2 + 19.4i)20-s + (−17.4 + 99.0i)22-s + (−116. − 97.6i)23-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.383 + 0.321i)4-s + (−0.117 + 0.667i)5-s + (−0.156 + 0.131i)7-s + (0.176 + 0.306i)8-s + (−0.239 + 0.414i)10-s + (0.239 + 1.35i)11-s + (−0.154 + 0.0563i)13-s + (−0.135 + 0.0494i)14-s + (0.0434 + 0.246i)16-s + (−0.249 + 0.431i)17-s + (0.438 + 0.759i)19-s + (−0.259 + 0.217i)20-s + (−0.169 + 0.959i)22-s + (−1.05 − 0.885i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.130 - 0.991i$
Analytic conductor: \(9.55830\)
Root analytic conductor: \(3.09165\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3/2),\ -0.130 - 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.46055 + 1.66471i\)
\(L(\frac12)\) \(\approx\) \(1.46055 + 1.66471i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.87 - 0.684i)T \)
3 \( 1 \)
good5 \( 1 + (1.31 - 7.46i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (2.90 - 2.43i)T + (59.5 - 337. i)T^{2} \)
11 \( 1 + (-8.73 - 49.5i)T + (-1.25e3 + 455. i)T^{2} \)
13 \( 1 + (7.25 - 2.64i)T + (1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (17.4 - 30.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-36.2 - 62.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (116. + 97.6i)T + (2.11e3 + 1.19e4i)T^{2} \)
29 \( 1 + (-60.4 - 21.9i)T + (1.86e4 + 1.56e4i)T^{2} \)
31 \( 1 + (-175. - 147. i)T + (5.17e3 + 2.93e4i)T^{2} \)
37 \( 1 + (-121. + 209. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (382. - 139. i)T + (5.27e4 - 4.43e4i)T^{2} \)
43 \( 1 + (62.2 + 353. i)T + (-7.47e4 + 2.71e4i)T^{2} \)
47 \( 1 + (-385. + 323. i)T + (1.80e4 - 1.02e5i)T^{2} \)
53 \( 1 - 70.4T + 1.48e5T^{2} \)
59 \( 1 + (-144. + 818. i)T + (-1.92e5 - 7.02e4i)T^{2} \)
61 \( 1 + (-540. + 453. i)T + (3.94e4 - 2.23e5i)T^{2} \)
67 \( 1 + (372. - 135. i)T + (2.30e5 - 1.93e5i)T^{2} \)
71 \( 1 + (-78.6 + 136. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (152. + 264. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-747. - 272. i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (-1.03e3 - 377. i)T + (4.38e5 + 3.67e5i)T^{2} \)
89 \( 1 + (-109. - 189. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (113. + 644. i)T + (-8.57e5 + 3.12e5i)T^{2} \)
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

−12.48738832623610107019853941349, −12.04401373638470257122547183066, −10.67857328495198000864560079547, −9.841418795828464564210020040469, −8.339850402700205672595491217941, −7.14291060343719356467499691199, −6.36935043493380480553042127261, −4.92119072763675633796498885684, −3.69219368435239298661422758891, −2.18691665446369288913515369268, 0.869194374566819123330474975041, 2.89652847036398545373358432780, 4.25390061147270037525643791694, 5.44073003980104525246779895482, 6.54996391779854571831174307371, 7.991846615202719694992863160478, 9.084955685235342762243559306976, 10.23069512030607257027380220460, 11.48753699334397311993769888974, 11.99604963067199773070765827196

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