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Properties

Label 2-2e5-32.5-c3-0-6
Degree $2$
Conductor $32$
Sign $-0.767 + 0.641i$
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  + (−2.82 − 0.161i)2-s + (−0.477 − 1.15i)3-s + (7.94 + 0.912i)4-s + (−16.3 − 6.77i)5-s + (1.16 + 3.33i)6-s + (−18.0 − 18.0i)7-s + (−22.2 − 3.85i)8-s + (17.9 − 17.9i)9-s + (45.0 + 21.7i)10-s + (−20.1 + 48.5i)11-s + (−2.74 − 9.60i)12-s + (37.8 − 15.6i)13-s + (47.9 + 53.8i)14-s + 22.0i·15-s + (62.3 + 14.4i)16-s − 53.0i·17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0571i)2-s + (−0.0919 − 0.221i)3-s + (0.993 + 0.114i)4-s + (−1.46 − 0.605i)5-s + (0.0791 + 0.226i)6-s + (−0.973 − 0.973i)7-s + (−0.985 − 0.170i)8-s + (0.666 − 0.666i)9-s + (1.42 + 0.688i)10-s + (−0.551 + 1.33i)11-s + (−0.0660 − 0.231i)12-s + (0.806 − 0.334i)13-s + (0.915 + 1.02i)14-s + 0.380i·15-s + (0.973 + 0.226i)16-s − 0.757i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.641i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.767 + 0.641i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ -0.767 + 0.641i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.129844 - 0.357892i\)
\(L(\frac12)\) \(\approx\) \(0.129844 - 0.357892i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.82 + 0.161i)T \)
good3 \( 1 + (0.477 + 1.15i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (16.3 + 6.77i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (18.0 + 18.0i)T + 343iT^{2} \)
11 \( 1 + (20.1 - 48.5i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-37.8 + 15.6i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 53.0iT - 4.91e3T^{2} \)
19 \( 1 + (32.4 - 13.4i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-32.1 + 32.1i)T - 1.21e4iT^{2} \)
29 \( 1 + (52.0 + 125. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 53.3T + 2.97e4T^{2} \)
37 \( 1 + (57.3 + 23.7i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-240. + 240. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-56.3 + 135. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 314. iT - 1.03e5T^{2} \)
53 \( 1 + (177. - 428. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (133. + 55.4i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (191. + 462. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (55.4 + 133. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-191. - 191. i)T + 3.57e5iT^{2} \)
73 \( 1 + (175. - 175. i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.22e3iT - 4.93e5T^{2} \)
83 \( 1 + (-896. + 371. i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-883. - 883. i)T + 7.04e5iT^{2} \)
97 \( 1 + 682.T + 9.12e5T^{2} \)
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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

−15.88168901531710874668938344225, −15.38405966265364520091182168913, −12.90507026151248993398676755449, −12.13047996794742563371058081681, −10.63698494838418754543869975453, −9.381723392096190393280877500936, −7.75466479456433893933781388147, −6.91117178757365194366590243776, −3.89194145272910869610367492157, −0.47555890881779977009707369367, 3.27234210240365563860886368596, 6.22283901929429207536359649488, 7.76264129899880622313953522439, 8.901329627451582878068100124279, 10.63552515752642799464561067057, 11.37786520550679927902044734391, 12.84242569121917376734357258684, 15.02617324440890694641010095806, 16.02185166604424841146257729786, 16.25239622209192899940335461957

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