↑

Properties

Label 2-2e5-32.11-c2-0-4
Degree $2$
Conductor $32$
Sign $0.776 + 0.629i$
Analytic cond. $0.871936$
Root an. cond. $0.933775$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.20 − 1.59i)2-s + (0.527 + 1.27i)3-s + (−1.09 − 3.84i)4-s + (−0.642 + 1.55i)5-s + (2.66 + 0.693i)6-s + (−4.95 + 4.95i)7-s + (−7.46 − 2.88i)8-s + (5.01 − 5.01i)9-s + (1.70 + 2.89i)10-s + (−4.27 + 10.3i)11-s + (4.32 − 3.42i)12-s + (1.68 + 4.06i)13-s + (1.93 + 13.8i)14-s − 2.31·15-s + (−13.6 + 8.42i)16-s − 28.6i·17-s + ⋯
L(s)  = 1  + (0.602 − 0.798i)2-s + (0.175 + 0.424i)3-s + (−0.273 − 0.961i)4-s + (−0.128 + 0.310i)5-s + (0.444 + 0.115i)6-s + (−0.707 + 0.707i)7-s + (−0.932 − 0.361i)8-s + (0.557 − 0.557i)9-s + (0.170 + 0.289i)10-s + (−0.388 + 0.937i)11-s + (0.360 − 0.285i)12-s + (0.129 + 0.312i)13-s + (0.138 + 0.990i)14-s − 0.154·15-s + (−0.850 + 0.526i)16-s − 1.68i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.776 + 0.629i$
Analytic conductor: \(0.871936\)
Root analytic conductor: \(0.933775\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1),\ 0.776 + 0.629i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.18161 - 0.418662i\)
\(L(\frac12)\) \(\approx\) \(1.18161 - 0.418662i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 + 1.59i)T \)
good3 \( 1 + (-0.527 - 1.27i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (0.642 - 1.55i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (4.95 - 4.95i)T - 49iT^{2} \)
11 \( 1 + (4.27 - 10.3i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-1.68 - 4.06i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + 28.6iT - 289T^{2} \)
19 \( 1 + (-17.5 + 7.26i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (24.3 + 24.3i)T + 529iT^{2} \)
29 \( 1 + (-8.57 + 3.55i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 5.73iT - 961T^{2} \)
37 \( 1 + (26.1 - 63.0i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (14.2 - 14.2i)T - 1.68e3iT^{2} \)
43 \( 1 + (10.1 - 24.4i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 57.9T + 2.20e3T^{2} \)
53 \( 1 + (46.3 + 19.2i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (27.6 + 11.4i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-76.3 + 31.6i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (36.1 + 87.3i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (5.39 - 5.39i)T - 5.04e3iT^{2} \)
73 \( 1 + (25.4 - 25.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 50.1T + 6.24e3T^{2} \)
83 \( 1 + (100. - 41.7i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-10.6 - 10.6i)T + 7.92e3iT^{2} \)
97 \( 1 + 14.3T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.93246100531235533602018333767, −15.26595759008843664379547329669, −13.99699530577898726259974533245, −12.63627857392056871960036076743, −11.71187076142217841415181269234, −10.05667933597423270631071711949, −9.307023671040865873212415837624, −6.72836307195201296567514565264, −4.79927998259821350863380309476, −2.97217843378945846819544581140, 3.76525843099119695129904537782, 5.78591632117481179358896580146, 7.33967419155833578597582302362, 8.434531865912031369606457067749, 10.42062745879756021111581514924, 12.36890280207669801235384349872, 13.29020619229742524100718469240, 14.08635194189433568345359915256, 15.79047619872664774645204276363, 16.37811137919586160220057168965

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