Properties

Label 2-2e7-16.3-c4-0-6
Degree $2$
Conductor $128$
Sign $0.999 - 0.0172i$
Analytic cond. $13.2313$
Root an. cond. $3.63749$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.5 + 11.5i)3-s + (14.6 − 14.6i)5-s − 24.0·7-s − 184. i·9-s + (−61.7 − 61.7i)11-s + (37.5 + 37.5i)13-s + 336. i·15-s + 96.8·17-s + (156. − 156. i)19-s + (276. − 276. i)21-s + 959.·23-s + 198. i·25-s + (1.19e3 + 1.19e3i)27-s + (350. + 350. i)29-s + 237. i·31-s + ⋯
L(s)  = 1  + (−1.28 + 1.28i)3-s + (0.584 − 0.584i)5-s − 0.490·7-s − 2.27i·9-s + (−0.510 − 0.510i)11-s + (0.222 + 0.222i)13-s + 1.49i·15-s + 0.335·17-s + (0.434 − 0.434i)19-s + (0.627 − 0.627i)21-s + 1.81·23-s + 0.317i·25-s + (1.63 + 1.63i)27-s + (0.416 + 0.416i)29-s + 0.247i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0172i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.0172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.999 - 0.0172i$
Analytic conductor: \(13.2313\)
Root analytic conductor: \(3.63749\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ 0.999 - 0.0172i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.06642 + 0.00920535i\)
\(L(\frac12)\) \(\approx\) \(1.06642 + 0.00920535i\)
\(L(3)\) 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 \( 1 + (11.5 - 11.5i)T - 81iT^{2} \)
5 \( 1 + (-14.6 + 14.6i)T - 625iT^{2} \)
7 \( 1 + 24.0T + 2.40e3T^{2} \)
11 \( 1 + (61.7 + 61.7i)T + 1.46e4iT^{2} \)
13 \( 1 + (-37.5 - 37.5i)T + 2.85e4iT^{2} \)
17 \( 1 - 96.8T + 8.35e4T^{2} \)
19 \( 1 + (-156. + 156. i)T - 1.30e5iT^{2} \)
23 \( 1 - 959.T + 2.79e5T^{2} \)
29 \( 1 + (-350. - 350. i)T + 7.07e5iT^{2} \)
31 \( 1 - 237. iT - 9.23e5T^{2} \)
37 \( 1 + (-560. + 560. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.80e3iT - 2.82e6T^{2} \)
43 \( 1 + (206. + 206. i)T + 3.41e6iT^{2} \)
47 \( 1 + 1.59e3iT - 4.87e6T^{2} \)
53 \( 1 + (-2.23e3 + 2.23e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (2.35e3 + 2.35e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-4.44e3 - 4.44e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (-3.99e3 + 3.99e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 4.92e3T + 2.54e7T^{2} \)
73 \( 1 + 2.65e3iT - 2.83e7T^{2} \)
79 \( 1 + 8.79e3iT - 3.89e7T^{2} \)
83 \( 1 + (-228. + 228. i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.05e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.10e4T + 8.85e7T^{2} \)
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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

−12.52076574160018122945468492173, −11.37040373281864445120338732964, −10.61052689522195354176817626583, −9.626176781833066043842889523106, −8.873794591610407043929343435131, −6.78474037707791052508830509329, −5.55549338604475805950574934221, −4.96404188294536172519443077561, −3.42270184442937235597246847437, −0.68203912932008945791613558230, 1.04156817342722921352837578780, 2.63435607914950669425236108286, 5.10529907286831376696229562578, 6.15811273619302662882004662466, 6.90447712732987908263251276457, 7.937434905333815808068485053710, 9.809838057000543767485396656739, 10.75330937741382913746996746568, 11.66980595068776117559673563926, 12.75937143954855668330638030802

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