Properties

Label 2-2e5-32.5-c1-0-2
Degree $2$
Conductor $32$
Sign $0.219 + 0.975i$
Analytic cond. $0.255521$
Root an. cond. $0.505491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.26 − 0.635i)2-s + (−1.07 − 2.60i)3-s + (1.19 + 1.60i)4-s + (0.707 + 0.292i)5-s + (−0.292 + 3.97i)6-s + (1.68 + 1.68i)7-s + (−0.484 − 2.78i)8-s + (−3.50 + 3.50i)9-s + (−0.707 − 0.819i)10-s + (−0.334 + 0.808i)11-s + (2.89 − 4.83i)12-s + (1.09 − 0.451i)13-s + (−1.05 − 3.20i)14-s − 2.15i·15-s + (−1.15 + 3.82i)16-s − 0.224i·17-s + ⋯
L(s)  = 1  + (−0.893 − 0.449i)2-s + (−0.623 − 1.50i)3-s + (0.595 + 0.803i)4-s + (0.316 + 0.130i)5-s + (−0.119 + 1.62i)6-s + (0.637 + 0.637i)7-s + (−0.171 − 0.985i)8-s + (−1.16 + 1.16i)9-s + (−0.223 − 0.259i)10-s + (−0.100 + 0.243i)11-s + (0.836 − 1.39i)12-s + (0.302 − 0.125i)13-s + (−0.282 − 0.855i)14-s − 0.557i·15-s + (−0.289 + 0.957i)16-s − 0.0545i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.219 + 0.975i$
Analytic conductor: \(0.255521\)
Root analytic conductor: \(0.505491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1/2),\ 0.219 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365599 - 0.292391i\)
\(L(\frac12)\) \(\approx\) \(0.365599 - 0.292391i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.635i)T \)
good3 \( 1 + (1.07 + 2.60i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.707 - 0.292i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.68 - 1.68i)T + 7iT^{2} \)
11 \( 1 + (0.334 - 0.808i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.09 + 0.451i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 0.224iT - 17T^{2} \)
19 \( 1 + (2.87 - 1.19i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.68 - 3.68i)T - 23iT^{2} \)
29 \( 1 + (-2.34 - 5.66i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + (9.87 + 4.09i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-6.37 + 6.37i)T - 41iT^{2} \)
43 \( 1 + (-1.90 + 4.60i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 0.542iT - 47T^{2} \)
53 \( 1 + (3.91 - 9.46i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.36 + 1.39i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.398 - 0.962i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-1.48 - 3.57i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (5.39 + 5.39i)T + 71iT^{2} \)
73 \( 1 + (5.15 - 5.15i)T - 73iT^{2} \)
79 \( 1 + 8.39iT - 79T^{2} \)
83 \( 1 + (-11.2 + 4.64i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.92 + 5.92i)T + 89iT^{2} \)
97 \( 1 + 4.19T + 97T^{2} \)
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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

−17.40897769943251374550937180755, −15.78028704778179298679005002876, −13.87399620616605496962080468136, −12.48711000067748710808361182242, −11.82818646825443887129531170663, −10.52658754860998971080776408403, −8.615721821083829182468653933697, −7.43248478488102221700541271466, −6.03542308789259743903214383662, −1.96593319095310802704015827057, 4.62719857476648949303599725849, 6.16041836783475513489727383420, 8.272089070586141363937390501569, 9.694712290614055803606118185768, 10.58014296224456791917416071205, 11.50712017753055219585144969468, 14.02146124115023427444711582202, 15.23824664159685127120234962326, 16.13338628512074037842460161629, 17.10245705881275001570732838183

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