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Properties

Label 2-2e5-32.3-c2-0-5
Degree $2$
Conductor $32$
Sign $-0.0865 + 0.996i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.44 − 1.38i)2-s + (0.936 − 2.26i)3-s + (0.175 + 3.99i)4-s + (−3.18 − 7.68i)5-s + (−4.47 + 1.97i)6-s + (3.67 + 3.67i)7-s + (5.27 − 6.01i)8-s + (2.12 + 2.12i)9-s + (−6.02 + 15.5i)10-s + (6.10 + 14.7i)11-s + (9.19 + 3.34i)12-s + (2.82 − 6.80i)13-s + (−0.228 − 10.3i)14-s − 20.3·15-s + (−15.9 + 1.40i)16-s − 3.67i·17-s + ⋯
L(s)  = 1  + (−0.722 − 0.691i)2-s + (0.312 − 0.753i)3-s + (0.0438 + 0.999i)4-s + (−0.636 − 1.53i)5-s + (−0.746 + 0.328i)6-s + (0.524 + 0.524i)7-s + (0.659 − 0.752i)8-s + (0.236 + 0.236i)9-s + (−0.602 + 1.55i)10-s + (0.554 + 1.33i)11-s + (0.766 + 0.278i)12-s + (0.216 − 0.523i)13-s + (−0.0162 − 0.742i)14-s − 1.35·15-s + (−0.996 + 0.0876i)16-s − 0.215i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0865 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0865 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.0865 + 0.996i$
Analytic conductor: \(0.871936\)
Root analytic conductor: \(0.933775\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1),\ -0.0865 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.524934 - 0.572497i\)
\(L(\frac12)\) \(\approx\) \(0.524934 - 0.572497i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.44 + 1.38i)T \)
good3 \( 1 + (-0.936 + 2.26i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (3.18 + 7.68i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (-3.67 - 3.67i)T + 49iT^{2} \)
11 \( 1 + (-6.10 - 14.7i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-2.82 + 6.80i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 3.67iT - 289T^{2} \)
19 \( 1 + (1.65 + 0.686i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (8.31 - 8.31i)T - 529iT^{2} \)
29 \( 1 + (38.8 + 16.0i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 4.11iT - 961T^{2} \)
37 \( 1 + (-19.8 - 47.9i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-21.1 - 21.1i)T + 1.68e3iT^{2} \)
43 \( 1 + (0.102 + 0.247i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 39.3T + 2.20e3T^{2} \)
53 \( 1 + (-22.6 + 9.36i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (101. - 41.9i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (14.0 + 5.81i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-3.67 + 8.87i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-75.7 - 75.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (29.0 + 29.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 2.76T + 6.24e3T^{2} \)
83 \( 1 + (79.1 + 32.8i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-72.4 + 72.4i)T - 7.92e3iT^{2} \)
97 \( 1 - 66.0T + 9.40e3T^{2} \)
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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

−16.58893933085540768437174364415, −15.32782022842986957666888222666, −13.23883724636955437420880399448, −12.47227709710950592610928955119, −11.62875921361047073809445527418, −9.625233916614435035789254969224, −8.403124929827612822797225347113, −7.54196096298141498350594402178, −4.54500816213483588616782769396, −1.62170832148395421389773165226, 3.84693200445681951709157948748, 6.38614068931851169833193961490, 7.66388780678388804145311632497, 9.144036888668808167156791654430, 10.63184348987180085492561064882, 11.22260457668720852179907644838, 14.14488219327480304980575596019, 14.64430706216752771302602774906, 15.71671200812563815547532030608, 16.71162638351974764329746013438

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