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Properties

Label 2-2e7-16.13-c11-0-41
Degree $2$
Conductor $128$
Sign $-0.958 - 0.285i$
Analytic cond. $98.3479$
Root an. cond. $9.91705$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.6 − 13.6i)3-s + (3.50e3 − 3.50e3i)5-s − 3.76e4i·7-s − 1.76e5i·9-s + (2.83e5 − 2.83e5i)11-s + (−1.70e6 − 1.70e6i)13-s − 9.52e4·15-s + 1.57e6·17-s + (−1.25e7 − 1.25e7i)19-s + (−5.12e5 + 5.12e5i)21-s − 6.83e6i·23-s + 2.43e7i·25-s + (−4.81e6 + 4.81e6i)27-s + (−5.00e7 − 5.00e7i)29-s + 7.19e7·31-s + ⋯
L(s)  = 1  + (−0.0323 − 0.0323i)3-s + (0.501 − 0.501i)5-s − 0.846i·7-s − 0.997i·9-s + (0.530 − 0.530i)11-s + (−1.27 − 1.27i)13-s − 0.0323·15-s + 0.269·17-s + (−1.15 − 1.15i)19-s + (−0.0273 + 0.0273i)21-s − 0.221i·23-s + 0.497i·25-s + (−0.0645 + 0.0645i)27-s + (−0.452 − 0.452i)29-s + 0.451·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.958 - 0.285i$
Analytic conductor: \(98.3479\)
Root analytic conductor: \(9.91705\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :11/2),\ -0.958 - 0.285i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.385841174\)
\(L(\frac12)\) \(\approx\) \(1.385841174\)
\(L(\frac{13}{2})\) 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 \( 1 + (13.6 + 13.6i)T + 1.77e5iT^{2} \)
5 \( 1 + (-3.50e3 + 3.50e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 3.76e4iT - 1.97e9T^{2} \)
11 \( 1 + (-2.83e5 + 2.83e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (1.70e6 + 1.70e6i)T + 1.79e12iT^{2} \)
17 \( 1 - 1.57e6T + 3.42e13T^{2} \)
19 \( 1 + (1.25e7 + 1.25e7i)T + 1.16e14iT^{2} \)
23 \( 1 + 6.83e6iT - 9.52e14T^{2} \)
29 \( 1 + (5.00e7 + 5.00e7i)T + 1.22e16iT^{2} \)
31 \( 1 - 7.19e7T + 2.54e16T^{2} \)
37 \( 1 + (9.81e7 - 9.81e7i)T - 1.77e17iT^{2} \)
41 \( 1 - 7.24e8iT - 5.50e17T^{2} \)
43 \( 1 + (1.93e8 - 1.93e8i)T - 9.29e17iT^{2} \)
47 \( 1 - 1.39e9T + 2.47e18T^{2} \)
53 \( 1 + (-3.32e9 + 3.32e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (5.46e9 - 5.46e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-4.88e9 - 4.88e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-7.47e9 - 7.47e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 2.58e10iT - 2.31e20T^{2} \)
73 \( 1 - 1.88e10iT - 3.13e20T^{2} \)
79 \( 1 - 3.09e10T + 7.47e20T^{2} \)
83 \( 1 + (2.96e10 + 2.96e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 6.42e10iT - 2.77e21T^{2} \)
97 \( 1 - 8.87e10T + 7.15e21T^{2} \)
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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

−10.52600347622193117984563488401, −9.638488765890274617872969642411, −8.672171061608147477498347915053, −7.38394850588428358151700631703, −6.33477777990493237851062159774, −5.15345846409101656840025583118, −3.94925244221479389084650678435, −2.66561207990530105285214816809, −1.03874057446377185301113927988, −0.31922222666736050673953332466, 1.90931211664677790026550316294, 2.34360881424234891853110772524, 4.13950153306311182135491232193, 5.29212928956093570171690074783, 6.42481496694261776096518317007, 7.46462892977659442795669149091, 8.763239662059909858994461819768, 9.776517221626124703070025011916, 10.62922276740695485252956326318, 11.90714084042480701980884998384

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