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Properties

Label 2-2e5-32.21-c1-0-0
Degree $2$
Conductor $32$
Sign $0.555 - 0.831i$
Analytic cond. $0.255521$
Root an. cond. $0.505491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−0.707 + 0.292i)3-s − 2.00·4-s + (1.12 − 2.70i)5-s + (−0.414 − 1.00i)6-s + (1 + i)7-s − 2.82i·8-s + (−1.70 + 1.70i)9-s + (3.82 + 1.58i)10-s + (−4.12 − 1.70i)11-s + (1.41 − 0.585i)12-s + (0.292 + 0.707i)13-s + (−1.41 + 1.41i)14-s + 2.24i·15-s + 4.00·16-s + 2.82i·17-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.408 + 0.169i)3-s − 1.00·4-s + (0.501 − 1.21i)5-s + (−0.169 − 0.408i)6-s + (0.377 + 0.377i)7-s − 1.00i·8-s + (−0.569 + 0.569i)9-s + (1.21 + 0.501i)10-s + (−1.24 − 0.514i)11-s + (0.408 − 0.169i)12-s + (0.0812 + 0.196i)13-s + (−0.377 + 0.377i)14-s + 0.579i·15-s + 1.00·16-s + 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(0.255521\)
Root analytic conductor: \(0.505491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1/2),\ 0.555 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.556099 + 0.297241i\)
\(L(\frac12)\) \(\approx\) \(0.556099 + 0.297241i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
good3 \( 1 + (0.707 - 0.292i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.12 + 2.70i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 + (4.12 + 1.70i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.292 - 0.707i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + (-1.53 - 3.70i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.82 + 5.82i)T - 23iT^{2} \)
29 \( 1 + (3.12 - 1.29i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-0.292 + 0.707i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.171 - 0.171i)T - 41iT^{2} \)
43 \( 1 + (-4.70 - 1.94i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 0.343iT - 47T^{2} \)
53 \( 1 + (1.12 + 0.464i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.87 - 4.53i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.70 + 0.707i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (5.53 - 2.29i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (5.82 + 5.82i)T + 71iT^{2} \)
73 \( 1 + (-7 + 7i)T - 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (-1.87 - 4.53i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-8.65 - 8.65i)T + 89iT^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−16.68195039181042013865371882985, −16.33197290303449701179261772819, −14.82744530218702008272726686027, −13.49979580423051699101090859473, −12.56606151155746504788626927166, −10.64958536266315894369863785049, −8.964605063623333690421383505347, −8.039681780751416313585515708752, −5.76679555433252604126019488759, −4.97583751447258677706480813545, 2.90836281020401295116138215067, 5.36197067505443972357916132126, 7.34456963808696468241255981512, 9.411335294844249942239170838702, 10.72033945482938012257679558430, 11.41255396244891181359750871590, 12.98852540977630713683271240559, 14.06162251805866850406576441488, 15.20225445524948300453701258012, 17.38104086478985457897226585608

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