↑

Properties

Label 2-2e5-32.29-c3-0-5
Degree $2$
Conductor $32$
Sign $0.395 - 0.918i$
Analytic cond. $1.88806$
Root an. cond. $1.37406$
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  + (1.17 + 2.57i)2-s + (7.57 + 3.13i)3-s + (−5.22 + 6.05i)4-s + (−8.03 − 19.4i)5-s + (0.844 + 23.1i)6-s + (−1.85 + 1.85i)7-s + (−21.7 − 6.32i)8-s + (28.3 + 28.3i)9-s + (40.4 − 43.5i)10-s + (8.63 − 3.57i)11-s + (−58.5 + 29.4i)12-s + (−11.5 + 27.9i)13-s + (−6.93 − 2.58i)14-s − 172. i·15-s + (−9.29 − 63.3i)16-s − 7.99i·17-s + ⋯
L(s)  = 1  + (0.416 + 0.909i)2-s + (1.45 + 0.603i)3-s + (−0.653 + 0.756i)4-s + (−0.718 − 1.73i)5-s + (0.0574 + 1.57i)6-s + (−0.0999 + 0.0999i)7-s + (−0.960 − 0.279i)8-s + (1.05 + 1.05i)9-s + (1.27 − 1.37i)10-s + (0.236 − 0.0979i)11-s + (−1.40 + 0.707i)12-s + (−0.247 + 0.597i)13-s + (−0.132 − 0.0492i)14-s − 2.96i·15-s + (−0.145 − 0.989i)16-s − 0.114i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.395 - 0.918i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.395 - 0.918i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.52028 + 1.00043i\)
\(L(\frac12)\) \(\approx\) \(1.52028 + 1.00043i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 - 2.57i)T \)
good3 \( 1 + (-7.57 - 3.13i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (8.03 + 19.4i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (1.85 - 1.85i)T - 343iT^{2} \)
11 \( 1 + (-8.63 + 3.57i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (11.5 - 27.9i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 7.99iT - 4.91e3T^{2} \)
19 \( 1 + (-5.76 + 13.9i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-60.2 - 60.2i)T + 1.21e4iT^{2} \)
29 \( 1 + (-167. - 69.3i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 225.T + 2.97e4T^{2} \)
37 \( 1 + (-0.431 - 1.04i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (275. + 275. i)T + 6.89e4iT^{2} \)
43 \( 1 + (-257. + 106. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 51.3iT - 1.03e5T^{2} \)
53 \( 1 + (2.98 - 1.23i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-101. - 244. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (270. + 111. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-778. - 322. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-484. + 484. i)T - 3.57e5iT^{2} \)
73 \( 1 + (212. + 212. i)T + 3.89e5iT^{2} \)
79 \( 1 + 593. iT - 4.93e5T^{2} \)
83 \( 1 + (-320. + 773. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-435. + 435. i)T - 7.04e5iT^{2} \)
97 \( 1 + 570.T + 9.12e5T^{2} \)
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

−16.11794100285676904627588744963, −15.49200328614128213330487761387, −14.28515467465189308963249721335, −13.23337121484621211728903596418, −12.12462321698778305725213156563, −9.237682105252411363914649952159, −8.783915345291148181353536776024, −7.59504536161378391696920565693, −4.94665738401804836568429794347, −3.77457883727207504260897840162, 2.58661485736058550420121867050, 3.61934487115080960383027731883, 6.81748616691275363812883213561, 8.191980276840900434330786687729, 9.889528498121141447559410477642, 11.11340150242056307482098554184, 12.53897219943770264230987195492, 13.83257193459346695911432441726, 14.64018123307421723179504876479, 15.25371219377940858655527268169

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