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Properties

Label 2-2e5-32.13-c3-0-8
Degree $2$
Conductor $32$
Sign $0.884 + 0.466i$
Analytic cond. $1.88806$
Root an. cond. $1.37406$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.81 − 0.280i)2-s + (1.64 − 3.96i)3-s + (7.84 − 1.57i)4-s + (−11.8 + 4.89i)5-s + (3.50 − 11.6i)6-s + (−5.11 + 5.11i)7-s + (21.6 − 6.63i)8-s + (6.09 + 6.09i)9-s + (−31.8 + 17.0i)10-s + (15.2 + 36.8i)11-s + (6.61 − 33.6i)12-s + (−73.4 − 30.4i)13-s + (−12.9 + 15.8i)14-s + 54.7i·15-s + (59.0 − 24.7i)16-s − 66.8i·17-s + ⋯
L(s)  = 1  + (0.995 − 0.0991i)2-s + (0.315 − 0.762i)3-s + (0.980 − 0.197i)4-s + (−1.05 + 0.437i)5-s + (0.238 − 0.789i)6-s + (−0.276 + 0.276i)7-s + (0.955 − 0.293i)8-s + (0.225 + 0.225i)9-s + (−1.00 + 0.540i)10-s + (0.417 + 1.00i)11-s + (0.159 − 0.809i)12-s + (−1.56 − 0.649i)13-s + (−0.247 + 0.302i)14-s + 0.943i·15-s + (0.922 − 0.386i)16-s − 0.954i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.884 + 0.466i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.90992 - 0.472863i\)
\(L(\frac12)\) \(\approx\) \(1.90992 - 0.472863i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.81 + 0.280i)T \)
good3 \( 1 + (-1.64 + 3.96i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (11.8 - 4.89i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (5.11 - 5.11i)T - 343iT^{2} \)
11 \( 1 + (-15.2 - 36.8i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (73.4 + 30.4i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 66.8iT - 4.91e3T^{2} \)
19 \( 1 + (37.0 + 15.3i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-30.1 - 30.1i)T + 1.21e4iT^{2} \)
29 \( 1 + (-64.4 + 155. i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 219.T + 2.97e4T^{2} \)
37 \( 1 + (286. - 118. i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-64.2 - 64.2i)T + 6.89e4iT^{2} \)
43 \( 1 + (200. + 484. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 392. iT - 1.03e5T^{2} \)
53 \( 1 + (-107. - 258. i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-237. + 98.4i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (43.9 - 106. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (333. - 804. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-387. + 387. i)T - 3.57e5iT^{2} \)
73 \( 1 + (518. + 518. i)T + 3.89e5iT^{2} \)
79 \( 1 + 214. iT - 4.93e5T^{2} \)
83 \( 1 + (-436. - 180. i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (877. - 877. i)T - 7.04e5iT^{2} \)
97 \( 1 - 43.7T + 9.12e5T^{2} \)
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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

−15.66750813427748837057806356862, −15.01794496589841203592596929608, −13.75352649598500718839321293036, −12.43878867171751677059413664579, −11.85991200297938774773000429485, −10.13155432245081028114259768233, −7.65523564732309586419858405096, −6.93700608178280934984476626546, −4.65835633230942450577799901185, −2.64138616358658204022883521442, 3.57292220300270735645941403625, 4.65467996357745991828672226241, 6.80434573757386671081981597388, 8.466380652692122544482425238567, 10.26219973268949794124303608573, 11.73858421839229739698491354748, 12.67154266234057107348496159424, 14.28279275232343453066859210743, 15.13491087627610083797055525936, 16.17006519507205379552710550809

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