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Properties

Label 2-2e5-32.13-c1-0-0
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: $\chi_{32} (13, \cdot )$
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

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

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