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Properties

Label 2-2e5-32.13-c3-0-4
Degree $2$
Conductor $32$
Sign $0.156 - 0.987i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.01 + 1.98i)2-s + (−1.20 + 2.90i)3-s + (0.125 + 7.99i)4-s + (−3.98 + 1.65i)5-s + (−8.17 + 3.46i)6-s + (22.4 − 22.4i)7-s + (−15.6 + 16.3i)8-s + (12.1 + 12.1i)9-s + (−11.3 − 4.58i)10-s + (−16.5 − 39.8i)11-s + (−23.3 − 9.25i)12-s + (17.9 + 7.42i)13-s + (89.6 − 0.701i)14-s − 13.5i·15-s + (−63.9 + 2.00i)16-s − 45.9i·17-s + ⋯
L(s)  = 1  + (0.712 + 0.701i)2-s + (−0.231 + 0.558i)3-s + (0.0156 + 0.999i)4-s + (−0.356 + 0.147i)5-s + (−0.556 + 0.235i)6-s + (1.20 − 1.20i)7-s + (−0.690 + 0.723i)8-s + (0.448 + 0.448i)9-s + (−0.357 − 0.144i)10-s + (−0.452 − 1.09i)11-s + (−0.561 − 0.222i)12-s + (0.382 + 0.158i)13-s + (1.71 − 0.0133i)14-s − 0.233i·15-s + (−0.999 + 0.0313i)16-s − 0.656i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.156 - 0.987i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.156 - 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.23195 + 1.05205i\)
\(L(\frac12)\) \(\approx\) \(1.23195 + 1.05205i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.01 - 1.98i)T \)
good3 \( 1 + (1.20 - 2.90i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (3.98 - 1.65i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (-22.4 + 22.4i)T - 343iT^{2} \)
11 \( 1 + (16.5 + 39.8i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (-17.9 - 7.42i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 45.9iT - 4.91e3T^{2} \)
19 \( 1 + (25.0 + 10.3i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (40.3 + 40.3i)T + 1.21e4iT^{2} \)
29 \( 1 + (88.6 - 214. i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 260.T + 2.97e4T^{2} \)
37 \( 1 + (-70.4 + 29.1i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (251. + 251. i)T + 6.89e4iT^{2} \)
43 \( 1 + (95.7 + 231. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 15.5iT - 1.03e5T^{2} \)
53 \( 1 + (-171. - 414. i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-53.3 + 22.0i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-297. + 718. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (377. - 911. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (359. - 359. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-605. - 605. i)T + 3.89e5iT^{2} \)
79 \( 1 + 380. iT - 4.93e5T^{2} \)
83 \( 1 + (-235. - 97.4i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (949. - 949. i)T - 7.04e5iT^{2} \)
97 \( 1 - 663.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.40326653535827299940483478152, −15.49352500117496073851212797417, −14.14731020684461459331792316653, −13.40708704045695023870476475901, −11.51072746603930541206885148610, −10.64177705935803586366460170137, −8.329013265857765032724034282943, −7.21436759672264583671127352423, −5.18393732027127625281554319255, −3.94903825119567701968918598362, 1.91182618904851289325809070503, 4.51554204127214950483426253881, 6.08269221208307949748252674566, 8.091333415648116702502630553151, 9.913951094816552687895469744416, 11.60814098192033750681836450079, 12.17920031962068268027156896671, 13.30313845370715016445545406107, 14.99587072961142188907717574430, 15.40566003445412610917851532160

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