Properties

Label 2-2e5-32.3-c2-0-0
Degree $2$
Conductor $32$
Sign $0.368 - 0.929i$
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  + (−1.96 − 0.384i)2-s + (−1.10 + 2.67i)3-s + (3.70 + 1.50i)4-s + (2.95 + 7.13i)5-s + (3.20 − 4.82i)6-s + (−4.18 − 4.18i)7-s + (−6.69 − 4.38i)8-s + (0.437 + 0.437i)9-s + (−3.05 − 15.1i)10-s + (1.42 + 3.44i)11-s + (−8.13 + 8.23i)12-s + (8.39 − 20.2i)13-s + (6.60 + 9.82i)14-s − 22.3·15-s + (11.4 + 11.1i)16-s + 1.73i·17-s + ⋯
L(s)  = 1  + (−0.981 − 0.192i)2-s + (−0.369 + 0.891i)3-s + (0.926 + 0.377i)4-s + (0.591 + 1.42i)5-s + (0.533 − 0.803i)6-s + (−0.597 − 0.597i)7-s + (−0.836 − 0.547i)8-s + (0.0486 + 0.0486i)9-s + (−0.305 − 1.51i)10-s + (0.129 + 0.313i)11-s + (−0.678 + 0.686i)12-s + (0.646 − 1.55i)13-s + (0.471 + 0.701i)14-s − 1.49·15-s + (0.715 + 0.698i)16-s + 0.101i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.368 - 0.929i$
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: \((2,\ 32,\ (\ :1),\ 0.368 - 0.929i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.538414 + 0.365704i\)
\(L(\frac12)\) \(\approx\) \(0.538414 + 0.365704i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 + 0.384i)T \)
good3 \( 1 + (1.10 - 2.67i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (-2.95 - 7.13i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (4.18 + 4.18i)T + 49iT^{2} \)
11 \( 1 + (-1.42 - 3.44i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-8.39 + 20.2i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 1.73iT - 289T^{2} \)
19 \( 1 + (-14.2 - 5.90i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-15.1 + 15.1i)T - 529iT^{2} \)
29 \( 1 + (6.74 + 2.79i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 + (-5.30 - 12.7i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (18.5 + 18.5i)T + 1.68e3iT^{2} \)
43 \( 1 + (-31.0 - 75.0i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 16.2T + 2.20e3T^{2} \)
53 \( 1 + (29.0 - 12.0i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-34.1 + 14.1i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (68.7 + 28.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-10.5 + 25.3i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (32.2 + 32.2i)T + 5.04e3iT^{2} \)
73 \( 1 + (28.5 + 28.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 22.4T + 6.24e3T^{2} \)
83 \( 1 + (123. + 51.0i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-61.0 + 61.0i)T - 7.92e3iT^{2} \)
97 \( 1 + 69.9T + 9.40e3T^{2} \)
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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

−16.94785903845440224944527015812, −15.81370320247183074366420526705, −14.87092447575565937491463033506, −13.11483173561296725178847355702, −11.10732771863122502140357155622, −10.37991085298396773578965333873, −9.719250319171774281156642125680, −7.55661083544069518792700062743, −6.14867038322074340446203340805, −3.23418827618785263716348558489, 1.40187753485139279992189654609, 5.70198562050249527448764124426, 6.93753023002268434798105583392, 8.806568014852890017662309142447, 9.454840873286585218640004588803, 11.56568717291843162517662250005, 12.50348740816206239312292494807, 13.71743788146355851047955659436, 15.78271057940164696650893934198, 16.58531626773983365378056467132

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