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Properties

Label 2-2e5-32.5-c3-0-2
Degree $2$
Conductor $32$
Sign $0.0173 - 0.999i$
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  + (1.17 + 2.57i)2-s + (0.729 + 1.76i)3-s + (−5.23 + 6.04i)4-s + (4.29 + 1.78i)5-s + (−3.67 + 3.94i)6-s + (1.47 + 1.47i)7-s + (−21.7 − 6.36i)8-s + (16.5 − 16.5i)9-s + (0.470 + 13.1i)10-s + (0.854 − 2.06i)11-s + (−14.4 − 4.81i)12-s + (40.9 − 16.9i)13-s + (−2.06 + 5.53i)14-s + 8.86i·15-s + (−9.13 − 63.3i)16-s − 73.1i·17-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (0.140 + 0.338i)3-s + (−0.654 + 0.755i)4-s + (0.384 + 0.159i)5-s + (−0.249 + 0.268i)6-s + (0.0798 + 0.0798i)7-s + (−0.959 − 0.281i)8-s + (0.611 − 0.611i)9-s + (0.0148 + 0.415i)10-s + (0.0234 − 0.0565i)11-s + (−0.348 − 0.115i)12-s + (0.874 − 0.362i)13-s + (−0.0394 + 0.105i)14-s + 0.152i·15-s + (−0.142 − 0.989i)16-s − 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0173 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0173 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.0173 - 0.999i$
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: \((2,\ 32,\ (\ :3/2),\ 0.0173 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.08814 + 1.06938i\)
\(L(\frac12)\) \(\approx\) \(1.08814 + 1.06938i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 - 2.57i)T \)
good3 \( 1 + (-0.729 - 1.76i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (-4.29 - 1.78i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (-1.47 - 1.47i)T + 343iT^{2} \)
11 \( 1 + (-0.854 + 2.06i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-40.9 + 16.9i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 73.1iT - 4.91e3T^{2} \)
19 \( 1 + (18.2 - 7.55i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (144. - 144. i)T - 1.21e4iT^{2} \)
29 \( 1 + (-80.7 - 194. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 168.T + 2.97e4T^{2} \)
37 \( 1 + (72.0 + 29.8i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (141. - 141. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-161. + 389. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 239. iT - 1.03e5T^{2} \)
53 \( 1 + (59.8 - 144. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-582. - 241. i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-238. - 575. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (156. + 376. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-411. - 411. i)T + 3.57e5iT^{2} \)
73 \( 1 + (642. - 642. i)T - 3.89e5iT^{2} \)
79 \( 1 + 800. iT - 4.93e5T^{2} \)
83 \( 1 + (1.34e3 - 557. i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-340. - 340. i)T + 7.04e5iT^{2} \)
97 \( 1 - 632.T + 9.12e5T^{2} \)
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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

−16.23337835825582234221938297512, −15.53745400670855542835965833010, −14.30801197153013944071119186420, −13.32171954328053912183173345671, −11.94573894551326074190134674762, −9.991621896183543254107338071137, −8.710848060192165020718785145820, −7.07682125586174631154120709868, −5.60155729400752022523476136511, −3.75434415993835785231920369833, 1.85620663281261435005816655094, 4.23396343518292752468246265581, 6.11892117388759631170662043014, 8.325092513470181679684200766352, 9.905472798290145511348114611894, 11.02643318337854803080926308128, 12.52046428270322759934919264758, 13.39911931968366007301723938147, 14.38549756817951132393268062241, 15.91423032750820844969002137192

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