Properties

Label 2-2e7-8.3-c2-0-5
Degree $2$
Conductor $128$
Sign $-0.707 + 0.707i$
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  − 2.82·3-s + 4i·5-s − 11.3i·7-s − 0.999·9-s − 14.1·11-s − 20i·13-s − 11.3i·15-s − 10·17-s − 14.1·19-s + 32.0i·21-s + 11.3i·23-s + 9·25-s + 28.2·27-s − 20i·29-s + 40.0·33-s + ⋯
L(s)  = 1  − 0.942·3-s + 0.800i·5-s − 1.61i·7-s − 0.111·9-s − 1.28·11-s − 1.53i·13-s − 0.754i·15-s − 0.588·17-s − 0.744·19-s + 1.52i·21-s + 0.491i·23-s + 0.359·25-s + 1.04·27-s − 0.689i·29-s + 1.21·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(3.48774\)
Root analytic conductor: \(1.86755\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.159683 - 0.385510i\)
\(L(\frac12)\) \(\approx\) \(0.159683 - 0.385510i\)
\(L(2)\) 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 \( 1 + 2.82T + 9T^{2} \)
5 \( 1 - 4iT - 25T^{2} \)
7 \( 1 + 11.3iT - 49T^{2} \)
11 \( 1 + 14.1T + 121T^{2} \)
13 \( 1 + 20iT - 169T^{2} \)
17 \( 1 + 10T + 289T^{2} \)
19 \( 1 + 14.1T + 361T^{2} \)
23 \( 1 - 11.3iT - 529T^{2} \)
29 \( 1 + 20iT - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 - 20iT - 1.36e3T^{2} \)
41 \( 1 + 30T + 1.68e3T^{2} \)
43 \( 1 + 2.82T + 1.84e3T^{2} \)
47 \( 1 - 67.8iT - 2.20e3T^{2} \)
53 \( 1 + 60iT - 2.80e3T^{2} \)
59 \( 1 - 42.4T + 3.48e3T^{2} \)
61 \( 1 - 28iT - 3.72e3T^{2} \)
67 \( 1 + 82.0T + 4.48e3T^{2} \)
71 \( 1 + 56.5iT - 5.04e3T^{2} \)
73 \( 1 + 10T + 5.32e3T^{2} \)
79 \( 1 + 113. iT - 6.24e3T^{2} \)
83 \( 1 + 25.4T + 6.88e3T^{2} \)
89 \( 1 - 22T + 7.92e3T^{2} \)
97 \( 1 - 150T + 9.40e3T^{2} \)
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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

−12.87015413378988659070923470927, −11.36213223011655804859577978871, −10.53684012740186773702093772102, −10.29486035259998602098819171307, −8.127285490425664775184963419116, −7.17861771775877453910480494413, −6.04789381260726145565317551499, −4.77190286585225553527462111598, −3.08585653525361006915529409156, −0.30186602861560070717357158788, 2.31198891437679015179785887199, 4.75566916414717388927805287607, 5.53213200004355704409551799839, 6.63552718198637616276331851219, 8.511336127630985663565392750028, 9.037620325160132971534567474899, 10.60042828047018257507524987421, 11.63860495340370683066650665553, 12.30929512421049125887639782999, 13.13615232413263606869495613649

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