↑

Properties

Label 2-2e5-32.29-c3-0-10
Degree $2$
Conductor $32$
Sign $-0.543 + 0.839i$
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  + (2.14 − 1.84i)2-s + (−9.18 − 3.80i)3-s + (1.20 − 7.90i)4-s + (1.04 + 2.51i)5-s + (−26.7 + 8.76i)6-s + (16.1 − 16.1i)7-s + (−11.9 − 19.1i)8-s + (50.8 + 50.8i)9-s + (6.87 + 3.47i)10-s + (3.72 − 1.54i)11-s + (−41.1 + 68.0i)12-s + (9.23 − 22.3i)13-s + (4.87 − 64.2i)14-s − 27.0i·15-s + (−61.0 − 19.0i)16-s − 4.95i·17-s + ⋯
L(s)  = 1  + (0.758 − 0.651i)2-s + (−1.76 − 0.732i)3-s + (0.150 − 0.988i)4-s + (0.0932 + 0.225i)5-s + (−1.81 + 0.596i)6-s + (0.869 − 0.869i)7-s + (−0.529 − 0.848i)8-s + (1.88 + 1.88i)9-s + (0.217 + 0.109i)10-s + (0.102 − 0.0422i)11-s + (−0.990 + 1.63i)12-s + (0.197 − 0.475i)13-s + (0.0930 − 1.22i)14-s − 0.466i·15-s + (−0.954 − 0.298i)16-s − 0.0706i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.543 + 0.839i$
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: \((2,\ 32,\ (\ :3/2),\ -0.543 + 0.839i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.544042 - 1.00073i\)
\(L(\frac12)\) \(\approx\) \(0.544042 - 1.00073i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.14 + 1.84i)T \)
good3 \( 1 + (9.18 + 3.80i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-1.04 - 2.51i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-16.1 + 16.1i)T - 343iT^{2} \)
11 \( 1 + (-3.72 + 1.54i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-9.23 + 22.3i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 4.95iT - 4.91e3T^{2} \)
19 \( 1 + (26.0 - 62.8i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-82.8 - 82.8i)T + 1.21e4iT^{2} \)
29 \( 1 + (-150. - 62.3i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 141.T + 2.97e4T^{2} \)
37 \( 1 + (1.05 + 2.55i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-8.70 - 8.70i)T + 6.89e4iT^{2} \)
43 \( 1 + (-290. + 120. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 450. iT - 1.03e5T^{2} \)
53 \( 1 + (114. - 47.4i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-124. - 300. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (223. + 92.7i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-204. - 84.5i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (606. - 606. i)T - 3.57e5iT^{2} \)
73 \( 1 + (531. + 531. i)T + 3.89e5iT^{2} \)
79 \( 1 + 1.12e3iT - 4.93e5T^{2} \)
83 \( 1 + (118. - 286. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-191. + 191. i)T - 7.04e5iT^{2} \)
97 \( 1 + 38.4T + 9.12e5T^{2} \)
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

−16.06657287996840612017534955422, −14.34409069899673897454523591888, −13.13589055155239255995698215895, −12.13486665054593137356507541447, −11.01341993613666658608758984997, −10.46627271943088740796803408838, −7.29382831261522449873595296977, −5.94506274946335279402240320202, −4.64470790840591105275538666923, −1.19398705804524267140774288096, 4.54253679672864215151648017994, 5.44781575991375893447426904268, 6.72728606579415435047411862595, 8.945743136221342244087464491094, 10.96676265693085182077435537364, 11.79903122078855090948124029526, 12.80570803961630572111688137055, 14.74870573615538341191224647305, 15.63724837698463051508069874114, 16.65197281068867875261127427382

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