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Properties

Label 2-2e5-32.5-c3-0-4
Degree $2$
Conductor $32$
Sign $0.971 + 0.238i$
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  + (1.49 − 2.39i)2-s + (2.92 + 7.06i)3-s + (−3.50 − 7.19i)4-s + (13.5 + 5.60i)5-s + (21.3 + 3.57i)6-s + (−23.0 − 23.0i)7-s + (−22.5 − 2.38i)8-s + (−22.2 + 22.2i)9-s + (33.7 − 24.0i)10-s + (−7.41 + 17.8i)11-s + (40.5 − 45.8i)12-s + (14.8 − 6.13i)13-s + (−89.9 + 20.7i)14-s + 111. i·15-s + (−39.4 + 50.3i)16-s − 27.7i·17-s + ⋯
L(s)  = 1  + (0.530 − 0.847i)2-s + (0.563 + 1.36i)3-s + (−0.437 − 0.899i)4-s + (1.20 + 0.501i)5-s + (1.45 + 0.243i)6-s + (−1.24 − 1.24i)7-s + (−0.994 − 0.105i)8-s + (−0.825 + 0.825i)9-s + (1.06 − 0.759i)10-s + (−0.203 + 0.490i)11-s + (0.976 − 1.10i)12-s + (0.316 − 0.130i)13-s + (−1.71 + 0.395i)14-s + 1.92i·15-s + (−0.616 + 0.787i)16-s − 0.396i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.971 + 0.238i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.971 + 0.238i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.76046 - 0.213453i\)
\(L(\frac12)\) \(\approx\) \(1.76046 - 0.213453i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.49 + 2.39i)T \)
good3 \( 1 + (-2.92 - 7.06i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (-13.5 - 5.60i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (23.0 + 23.0i)T + 343iT^{2} \)
11 \( 1 + (7.41 - 17.8i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-14.8 + 6.13i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 27.7iT - 4.91e3T^{2} \)
19 \( 1 + (82.2 - 34.0i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-39.2 + 39.2i)T - 1.21e4iT^{2} \)
29 \( 1 + (-5.57 - 13.4i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 155.T + 2.97e4T^{2} \)
37 \( 1 + (-188. - 78.1i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-113. + 113. i)T - 6.89e4iT^{2} \)
43 \( 1 + (24.6 - 59.5i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 217. iT - 1.03e5T^{2} \)
53 \( 1 + (35.7 - 86.3i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (116. + 48.2i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (197. + 475. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-144. - 349. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (523. + 523. i)T + 3.57e5iT^{2} \)
73 \( 1 + (718. - 718. i)T - 3.89e5iT^{2} \)
79 \( 1 + 958. iT - 4.93e5T^{2} \)
83 \( 1 + (-1.24e3 + 514. i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (808. + 808. i)T + 7.04e5iT^{2} \)
97 \( 1 - 1.39e3T + 9.12e5T^{2} \)
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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

−16.05696900894972831467629539162, −14.77386340279689221055993358061, −13.84629640534730038761892884209, −12.94937243089406501102602023780, −10.62760338741284988188352381123, −10.13222577113726521823277881472, −9.331599336989624512753184958724, −6.31895173480101928989961616455, −4.36325576724310618641798518792, −2.96031447570338031058726118505, 2.61118780324408515356595023412, 5.80571953952951892363616050657, 6.56610821208589382990941427181, 8.408207780688359025026779251895, 9.299795763137190111498711213817, 12.26723224738522254139469109130, 13.13853194253254142221927118434, 13.53894045900059662035567454075, 15.01844905622095393907287138891, 16.34038238938742563546190491518

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