Properties

Label 2-2e7-16.11-c2-0-3
Degree $2$
Conductor $128$
Sign $0.631 + 0.775i$
Analytic cond. $3.48774$
Root an. cond. $1.86755$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.146 + 0.146i)3-s + (−3.68 − 3.68i)5-s + 9.66·7-s − 8.95i·9-s + (5.51 − 5.51i)11-s + (6.27 − 6.27i)13-s − 1.07i·15-s − 6.78·17-s + (13.5 + 13.5i)19-s + (1.41 + 1.41i)21-s − 17.0·23-s + 2.17i·25-s + (2.62 − 2.62i)27-s + (−4.85 + 4.85i)29-s − 5.25i·31-s + ⋯
L(s)  = 1  + (0.0487 + 0.0487i)3-s + (−0.737 − 0.737i)5-s + 1.38·7-s − 0.995i·9-s + (0.501 − 0.501i)11-s + (0.482 − 0.482i)13-s − 0.0719i·15-s − 0.399·17-s + (0.711 + 0.711i)19-s + (0.0673 + 0.0673i)21-s − 0.742·23-s + 0.0868i·25-s + (0.0973 − 0.0973i)27-s + (−0.167 + 0.167i)29-s − 0.169i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.631 + 0.775i$
Analytic conductor: \(3.48774\)
Root analytic conductor: \(1.86755\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.631 + 0.775i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.28483 - 0.610715i\)
\(L(\frac12)\) \(\approx\) \(1.28483 - 0.610715i\)
\(L(2)\) 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 \( 1 + (-0.146 - 0.146i)T + 9iT^{2} \)
5 \( 1 + (3.68 + 3.68i)T + 25iT^{2} \)
7 \( 1 - 9.66T + 49T^{2} \)
11 \( 1 + (-5.51 + 5.51i)T - 121iT^{2} \)
13 \( 1 + (-6.27 + 6.27i)T - 169iT^{2} \)
17 \( 1 + 6.78T + 289T^{2} \)
19 \( 1 + (-13.5 - 13.5i)T + 361iT^{2} \)
23 \( 1 + 17.0T + 529T^{2} \)
29 \( 1 + (4.85 - 4.85i)T - 841iT^{2} \)
31 \( 1 + 5.25iT - 961T^{2} \)
37 \( 1 + (-18.1 - 18.1i)T + 1.36e3iT^{2} \)
41 \( 1 - 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (54.5 - 54.5i)T - 1.84e3iT^{2} \)
47 \( 1 - 40.4iT - 2.20e3T^{2} \)
53 \( 1 + (10.8 + 10.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (-50.8 + 50.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (-17.0 + 17.0i)T - 3.72e3iT^{2} \)
67 \( 1 + (-22.9 - 22.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 51.6T + 5.04e3T^{2} \)
73 \( 1 + 78.5iT - 5.32e3T^{2} \)
79 \( 1 - 108. iT - 6.24e3T^{2} \)
83 \( 1 + (-57.3 - 57.3i)T + 6.88e3iT^{2} \)
89 \( 1 + 44.1iT - 7.92e3T^{2} \)
97 \( 1 - 112.T + 9.40e3T^{2} \)
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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

−12.80978142334865857411281985997, −11.76212633651133948346070175260, −11.28352406155003716726192569975, −9.704769410609953027441049197993, −8.453863103014025218110663542876, −7.943865604055658519035849366271, −6.23294661261759384540442188834, −4.81195408157694058549468912775, −3.66162520057796070896004407721, −1.14666983541597035126335982429, 2.02279808850000456988517517984, 3.97927758473003112946790116109, 5.17522856817213202663390405637, 6.97906298916505510026264133683, 7.77843680313576712498394008786, 8.828233926513025312699682303178, 10.42423034135515169326788489772, 11.32239588029299951092380265210, 11.83512446379618725767202580789, 13.48199774613989454729768346724

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