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Properties

Label 2-2e5-32.29-c3-0-3
Degree $2$
Conductor $32$
Sign $0.978 + 0.206i$
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  + (−1.99 − 2.00i)2-s + (6.97 + 2.89i)3-s + (−0.0381 + 7.99i)4-s + (1.57 + 3.79i)5-s + (−8.13 − 19.7i)6-s + (15.6 − 15.6i)7-s + (16.1 − 15.8i)8-s + (21.2 + 21.2i)9-s + (4.47 − 10.7i)10-s + (−56.0 + 23.2i)11-s + (−23.3 + 55.7i)12-s + (14.0 − 34.0i)13-s + (−62.6 − 0.149i)14-s + 31.0i·15-s + (−63.9 − 0.609i)16-s + 26.6i·17-s + ⋯
L(s)  = 1  + (−0.705 − 0.708i)2-s + (1.34 + 0.556i)3-s + (−0.00476 + 0.999i)4-s + (0.140 + 0.339i)5-s + (−0.553 − 1.34i)6-s + (0.845 − 0.845i)7-s + (0.712 − 0.702i)8-s + (0.787 + 0.787i)9-s + (0.141 − 0.339i)10-s + (−1.53 + 0.636i)11-s + (−0.562 + 1.34i)12-s + (0.300 − 0.726i)13-s + (−1.19 − 0.00284i)14-s + 0.534i·15-s + (−0.999 − 0.00953i)16-s + 0.379i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.978 + 0.206i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.978 + 0.206i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.28943 - 0.134758i\)
\(L(\frac12)\) \(\approx\) \(1.28943 - 0.134758i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 + 2.00i)T \)
good3 \( 1 + (-6.97 - 2.89i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-1.57 - 3.79i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-15.6 + 15.6i)T - 343iT^{2} \)
11 \( 1 + (56.0 - 23.2i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-14.0 + 34.0i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 26.6iT - 4.91e3T^{2} \)
19 \( 1 + (30.7 - 74.2i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (141. + 141. i)T + 1.21e4iT^{2} \)
29 \( 1 + (11.9 + 4.97i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 128.T + 2.97e4T^{2} \)
37 \( 1 + (-85.1 - 205. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (32.3 + 32.3i)T + 6.89e4iT^{2} \)
43 \( 1 + (-314. + 130. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 184. iT - 1.03e5T^{2} \)
53 \( 1 + (-277. + 114. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-241. - 582. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-297. - 123. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (605. + 250. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-163. + 163. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-624. - 624. i)T + 3.89e5iT^{2} \)
79 \( 1 + 139. iT - 4.93e5T^{2} \)
83 \( 1 + (226. - 545. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-231. + 231. i)T - 7.04e5iT^{2} \)
97 \( 1 + 594.T + 9.12e5T^{2} \)
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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

−16.29648579754816228368736369372, −14.99428583916024316036190854980, −13.90194479447202028875845515176, −12.67116753445581937602503261986, −10.56792148846703857760192808809, −10.18509089828979532810543673240, −8.387522750218622658278541039651, −7.73310490203826225941193272418, −4.15480137700629904404432808361, −2.45082220122264867143040052679, 2.13778435116913086384635006826, 5.42937155758148882448913605197, 7.51182063131535983447646149948, 8.441972234997223629657412652279, 9.272965493268414297103130198658, 11.17260423417472669324200524915, 13.19915510399505861463142300521, 14.11309856534669156198612114322, 15.19496796336570155146261139266, 16.15510085400653683394919908384

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