Properties

Label 2-2e5-32.13-c7-0-3
Degree $2$
Conductor $32$
Sign $-0.262 + 0.965i$
Analytic cond. $9.99632$
Root an. cond. $3.16169$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 + 11.1i)2-s + (−17.7 + 42.9i)3-s + (−122. + 36.1i)4-s + (97.0 − 40.1i)5-s + (−509. − 129. i)6-s + (−891. + 891. i)7-s + (−603. − 1.31e3i)8-s + (21.3 + 21.3i)9-s + (606. + 1.02e3i)10-s + (−2.87e3 − 6.95e3i)11-s + (629. − 5.91e3i)12-s + (8.85e3 + 3.66e3i)13-s + (−1.14e4 − 8.54e3i)14-s + 4.87e3i·15-s + (1.37e4 − 8.88e3i)16-s + 2.20e3i·17-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)2-s + (−0.380 + 0.917i)3-s + (−0.959 + 0.282i)4-s + (0.347 − 0.143i)5-s + (−0.962 − 0.245i)6-s + (−0.982 + 0.982i)7-s + (−0.416 − 0.909i)8-s + (0.00975 + 0.00975i)9-s + (0.191 + 0.322i)10-s + (−0.652 − 1.57i)11-s + (0.105 − 0.987i)12-s + (1.11 + 0.463i)13-s + (−1.11 − 0.832i)14-s + 0.373i·15-s + (0.840 − 0.542i)16-s + 0.108i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.965i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.262 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.262 + 0.965i$
Analytic conductor: \(9.99632\)
Root analytic conductor: \(3.16169\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :7/2),\ -0.262 + 0.965i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.340876 - 0.445866i\)
\(L(\frac12)\) \(\approx\) \(0.340876 - 0.445866i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.61 - 11.1i)T \)
good3 \( 1 + (17.7 - 42.9i)T + (-1.54e3 - 1.54e3i)T^{2} \)
5 \( 1 + (-97.0 + 40.1i)T + (5.52e4 - 5.52e4i)T^{2} \)
7 \( 1 + (891. - 891. i)T - 8.23e5iT^{2} \)
11 \( 1 + (2.87e3 + 6.95e3i)T + (-1.37e7 + 1.37e7i)T^{2} \)
13 \( 1 + (-8.85e3 - 3.66e3i)T + (4.43e7 + 4.43e7i)T^{2} \)
17 \( 1 - 2.20e3iT - 4.10e8T^{2} \)
19 \( 1 + (1.36e4 + 5.63e3i)T + (6.32e8 + 6.32e8i)T^{2} \)
23 \( 1 + (4.12e4 + 4.12e4i)T + 3.40e9iT^{2} \)
29 \( 1 + (6.09e4 - 1.47e5i)T + (-1.21e10 - 1.21e10i)T^{2} \)
31 \( 1 + 8.92e4T + 2.75e10T^{2} \)
37 \( 1 + (2.00e5 - 8.29e4i)T + (6.71e10 - 6.71e10i)T^{2} \)
41 \( 1 + (1.39e5 + 1.39e5i)T + 1.94e11iT^{2} \)
43 \( 1 + (-3.80e5 - 9.19e5i)T + (-1.92e11 + 1.92e11i)T^{2} \)
47 \( 1 + 7.01e5iT - 5.06e11T^{2} \)
53 \( 1 + (5.73e4 + 1.38e5i)T + (-8.30e11 + 8.30e11i)T^{2} \)
59 \( 1 + (1.91e6 - 7.94e5i)T + (1.75e12 - 1.75e12i)T^{2} \)
61 \( 1 + (-8.51e5 + 2.05e6i)T + (-2.22e12 - 2.22e12i)T^{2} \)
67 \( 1 + (3.55e5 - 8.59e5i)T + (-4.28e12 - 4.28e12i)T^{2} \)
71 \( 1 + (-1.33e6 + 1.33e6i)T - 9.09e12iT^{2} \)
73 \( 1 + (-1.66e6 - 1.66e6i)T + 1.10e13iT^{2} \)
79 \( 1 - 6.31e6iT - 1.92e13T^{2} \)
83 \( 1 + (-6.87e6 - 2.84e6i)T + (1.91e13 + 1.91e13i)T^{2} \)
89 \( 1 + (4.41e6 - 4.41e6i)T - 4.42e13iT^{2} \)
97 \( 1 + 6.87e6T + 8.07e13T^{2} \)
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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

−16.15387190066609325939791354923, −15.37803492318533970595077803411, −13.78554611066919944995122413894, −12.80168380162968694363492887491, −10.91559145792446506182722938534, −9.464269943894095934895417315584, −8.486135399986382439111979219895, −6.25036578075609214534981365925, −5.40694662228526950389573351203, −3.58068065025669621423319370908, 0.26039739968081571482901838047, 1.88189346633891384157728305810, 3.91384869793117504663245089677, 6.04946957463710230603279399023, 7.54514812431229019114429934668, 9.689265135416045892205253943499, 10.54659096776277993190222428308, 12.16643243362265297378304532675, 13.04516456868343974978013656302, 13.70929979871453077388036604306

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