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Properties

Label 2-2e7-16.3-c12-0-45
Degree $2$
Conductor $128$
Sign $-0.897 - 0.441i$
Analytic cond. $116.991$
Root an. cond. $10.8162$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (980. − 980. i)3-s + (1.59e4 − 1.59e4i)5-s − 9.48e4·7-s − 1.39e6i·9-s + (−4.64e5 − 4.64e5i)11-s + (−1.28e6 − 1.28e6i)13-s − 3.11e7i·15-s − 2.93e7·17-s + (3.66e7 − 3.66e7i)19-s + (−9.29e7 + 9.29e7i)21-s + 1.23e8·23-s − 2.61e8i·25-s + (−8.43e8 − 8.43e8i)27-s + (2.94e8 + 2.94e8i)29-s + 9.22e8i·31-s + ⋯
L(s)  = 1  + (1.34 − 1.34i)3-s + (1.01 − 1.01i)5-s − 0.806·7-s − 2.61i·9-s + (−0.261 − 0.261i)11-s + (−0.265 − 0.265i)13-s − 2.73i·15-s − 1.21·17-s + (0.779 − 0.779i)19-s + (−1.08 + 1.08i)21-s + 0.832·23-s − 1.07i·25-s + (−2.17 − 2.17i)27-s + (0.495 + 0.495i)29-s + 1.03i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.897 - 0.441i$
Analytic conductor: \(116.991\)
Root analytic conductor: \(10.8162\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :6),\ -0.897 - 0.441i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(3.137432431\)
\(L(\frac12)\) \(\approx\) \(3.137432431\)
\(L(7)\) 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 \( 1 + (-980. + 980. i)T - 5.31e5iT^{2} \)
5 \( 1 + (-1.59e4 + 1.59e4i)T - 2.44e8iT^{2} \)
7 \( 1 + 9.48e4T + 1.38e10T^{2} \)
11 \( 1 + (4.64e5 + 4.64e5i)T + 3.13e12iT^{2} \)
13 \( 1 + (1.28e6 + 1.28e6i)T + 2.32e13iT^{2} \)
17 \( 1 + 2.93e7T + 5.82e14T^{2} \)
19 \( 1 + (-3.66e7 + 3.66e7i)T - 2.21e15iT^{2} \)
23 \( 1 - 1.23e8T + 2.19e16T^{2} \)
29 \( 1 + (-2.94e8 - 2.94e8i)T + 3.53e17iT^{2} \)
31 \( 1 - 9.22e8iT - 7.87e17T^{2} \)
37 \( 1 + (1.29e9 - 1.29e9i)T - 6.58e18iT^{2} \)
41 \( 1 + 8.03e8iT - 2.25e19T^{2} \)
43 \( 1 + (4.51e9 + 4.51e9i)T + 3.99e19iT^{2} \)
47 \( 1 + 3.52e9iT - 1.16e20T^{2} \)
53 \( 1 + (-1.65e10 + 1.65e10i)T - 4.91e20iT^{2} \)
59 \( 1 + (-4.94e9 - 4.94e9i)T + 1.77e21iT^{2} \)
61 \( 1 + (-5.20e10 - 5.20e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (7.31e10 - 7.31e10i)T - 8.18e21iT^{2} \)
71 \( 1 + 3.30e10T + 1.64e22T^{2} \)
73 \( 1 + 6.47e10iT - 2.29e22T^{2} \)
79 \( 1 + 1.41e11iT - 5.90e22T^{2} \)
83 \( 1 + (-2.72e11 + 2.72e11i)T - 1.06e23iT^{2} \)
89 \( 1 - 5.11e11iT - 2.46e23T^{2} \)
97 \( 1 - 1.62e12T + 6.93e23T^{2} \)
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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

−10.05039528730539322985754511529, −8.914337502347752442075051938786, −8.732717996078133362555192528250, −7.21899286918091873006877117644, −6.46880779584038173851805532980, −5.10864121091703626992472526934, −3.27760960728537035936522164174, −2.40932439185169428065113332646, −1.38868246273016716655890463663, −0.47535025620062071584082050026, 2.09923283503057882166428704139, 2.77000737130227889563366249527, 3.67829088728586893755243423652, 4.91707132526417226697686254720, 6.30771508647618374585510776153, 7.54779985074099110263385400343, 8.902223272165962384986334808744, 9.728266351935008298330130968630, 10.15287233648081633290696581713, 11.14915842330542178913617940142

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