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Properties

Label 2-2e7-32.5-c3-0-8
Degree $2$
Conductor $128$
Sign $0.172 + 0.985i$
Analytic cond. $7.55224$
Root an. cond. $2.74813$
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  + (1.20 + 2.90i)3-s + (−3.98 − 1.65i)5-s + (−22.4 − 22.4i)7-s + (12.1 − 12.1i)9-s + (16.5 − 39.8i)11-s + (17.9 − 7.42i)13-s − 13.5i·15-s + 45.9i·17-s + (25.0 − 10.3i)19-s + (38.0 − 91.9i)21-s + (40.3 − 40.3i)23-s + (−75.2 − 75.2i)25-s + (128. + 53.0i)27-s + (−88.6 − 214. i)29-s − 260.·31-s + ⋯
L(s)  = 1  + (0.231 + 0.558i)3-s + (−0.356 − 0.147i)5-s + (−1.20 − 1.20i)7-s + (0.448 − 0.448i)9-s + (0.452 − 1.09i)11-s + (0.382 − 0.158i)13-s − 0.233i·15-s + 0.656i·17-s + (0.301 − 0.125i)19-s + (0.395 − 0.955i)21-s + (0.365 − 0.365i)23-s + (−0.601 − 0.601i)25-s + (0.912 + 0.378i)27-s + (−0.567 − 1.37i)29-s − 1.50·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.172 + 0.985i$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.172 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.949551 - 0.798121i\)
\(L(\frac12)\) \(\approx\) \(0.949551 - 0.798121i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-1.20 - 2.90i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (3.98 + 1.65i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (22.4 + 22.4i)T + 343iT^{2} \)
11 \( 1 + (-16.5 + 39.8i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-17.9 + 7.42i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 45.9iT - 4.91e3T^{2} \)
19 \( 1 + (-25.0 + 10.3i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-40.3 + 40.3i)T - 1.21e4iT^{2} \)
29 \( 1 + (88.6 + 214. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 260.T + 2.97e4T^{2} \)
37 \( 1 + (-70.4 - 29.1i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (251. - 251. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-95.7 + 231. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 15.5iT - 1.03e5T^{2} \)
53 \( 1 + (-171. + 414. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (53.3 + 22.0i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-297. - 718. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-377. - 911. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-359. - 359. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-605. + 605. i)T - 3.89e5iT^{2} \)
79 \( 1 + 380. iT - 4.93e5T^{2} \)
83 \( 1 + (235. - 97.4i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (949. + 949. i)T + 7.04e5iT^{2} \)
97 \( 1 - 663.T + 9.12e5T^{2} \)
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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

−12.86325073465290398929037183718, −11.49532014145400150848775266144, −10.39706182742349861396994886838, −9.623654762882770014195775037313, −8.501507176835874872514395833707, −7.11189620787307180930228987821, −6.05549477611154861910933596205, −4.05012928506590247765781441997, −3.49379541738420790682590524318, −0.63228437548960960805254311552, 1.97025105960236915606337586057, 3.48218512938854655583424330218, 5.29191708182411639751134101308, 6.72521050960980811503263634706, 7.50353649020776422831737511061, 9.017652208088616528089557307805, 9.705461193936817609345658223927, 11.20193436707677956826926614793, 12.41513961067491814484456741687, 12.79716982511781770965148416724

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