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Properties

Label 2-2e5-32.29-c1-0-2
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: $\chi_{32} (29, \cdot )$
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

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

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