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Properties

Label 2-2e5-32.29-c3-0-0
Degree $2$
Conductor $32$
Sign $-0.0675 - 0.997i$
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  + (−2.58 − 1.14i)2-s + (−5.56 − 2.30i)3-s + (5.35 + 5.94i)4-s + (6.28 + 15.1i)5-s + (11.7 + 12.3i)6-s + (−16.6 + 16.6i)7-s + (−7.02 − 21.5i)8-s + (6.60 + 6.60i)9-s + (1.19 − 46.4i)10-s + (3.11 − 1.28i)11-s + (−16.1 − 45.4i)12-s + (−28.8 + 69.7i)13-s + (62.1 − 23.9i)14-s − 98.9i·15-s + (−6.57 + 63.6i)16-s − 66.2i·17-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (−1.07 − 0.443i)3-s + (0.669 + 0.742i)4-s + (0.562 + 1.35i)5-s + (0.798 + 0.841i)6-s + (−0.899 + 0.899i)7-s + (−0.310 − 0.950i)8-s + (0.244 + 0.244i)9-s + (0.0378 − 1.46i)10-s + (0.0853 − 0.0353i)11-s + (−0.388 − 1.09i)12-s + (−0.616 + 1.48i)13-s + (1.18 − 0.456i)14-s − 1.70i·15-s + (−0.102 + 0.994i)16-s − 0.944i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0675 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0675 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.0675 - 0.997i$
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: \((2,\ 32,\ (\ :3/2),\ -0.0675 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.279484 + 0.299052i\)
\(L(\frac12)\) \(\approx\) \(0.279484 + 0.299052i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.58 + 1.14i)T \)
good3 \( 1 + (5.56 + 2.30i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-6.28 - 15.1i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (16.6 - 16.6i)T - 343iT^{2} \)
11 \( 1 + (-3.11 + 1.28i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (28.8 - 69.7i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 66.2iT - 4.91e3T^{2} \)
19 \( 1 + (12.5 - 30.2i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (63.7 + 63.7i)T + 1.21e4iT^{2} \)
29 \( 1 + (-190. - 79.0i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 123.T + 2.97e4T^{2} \)
37 \( 1 + (46.0 + 111. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (100. + 100. i)T + 6.89e4iT^{2} \)
43 \( 1 + (-27.5 + 11.4i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 394. iT - 1.03e5T^{2} \)
53 \( 1 + (-135. + 56.0i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-297. - 717. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (548. + 227. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-163. - 67.6i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-194. + 194. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-547. - 547. i)T + 3.89e5iT^{2} \)
79 \( 1 - 715. iT - 4.93e5T^{2} \)
83 \( 1 + (-54.6 + 131. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-220. + 220. i)T - 7.04e5iT^{2} \)
97 \( 1 - 1.36e3T + 9.12e5T^{2} \)
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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

−16.93437013210689268226107475629, −15.88620827128411089611456973656, −14.21376215840643155876671061093, −12.34354248412924736066389495344, −11.63257285040211248085100419401, −10.36561505925168595109828836612, −9.235920667010862843645148683544, −6.91100024389589114025628247705, −6.27594806014611593905429482772, −2.59785318531544248620307171477, 0.54646206385723969630686662837, 5.07910840853422247328651475456, 6.27525712609495672268117744530, 8.199260236396395370744031860985, 9.841356099189015605302094136549, 10.44186313176056717721385336039, 12.15373597546187358334948435311, 13.44359336959243269346988454866, 15.47642041260745623710732032385, 16.42480611913736485951488044731

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