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Properties

Label 2-2e7-8.3-c4-0-9
Degree $2$
Conductor $128$
Sign $0.707 - 0.707i$
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  + 15.8·3-s + 40.8i·5-s − 40.7i·7-s + 170.·9-s + 42.9·11-s + 186. i·13-s + 647. i·15-s − 157.·17-s + 278.·19-s − 646. i·21-s − 249. i·23-s − 1.04e3·25-s + 1.41e3·27-s − 416. i·29-s + 1.08e3i·31-s + ⋯
L(s)  = 1  + 1.76·3-s + 1.63i·5-s − 0.832i·7-s + 2.10·9-s + 0.354·11-s + 1.10i·13-s + 2.87i·15-s − 0.544·17-s + 0.770·19-s − 1.46i·21-s − 0.472i·23-s − 1.66·25-s + 1.94·27-s − 0.495i·29-s + 1.13i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(13.2313\)
Root analytic conductor: \(3.63749\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.99937 + 1.24238i\)
\(L(\frac12)\) \(\approx\) \(2.99937 + 1.24238i\)
\(L(3)\) 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 \( 1 - 15.8T + 81T^{2} \)
5 \( 1 - 40.8iT - 625T^{2} \)
7 \( 1 + 40.7iT - 2.40e3T^{2} \)
11 \( 1 - 42.9T + 1.46e4T^{2} \)
13 \( 1 - 186. iT - 2.85e4T^{2} \)
17 \( 1 + 157.T + 8.35e4T^{2} \)
19 \( 1 - 278.T + 1.30e5T^{2} \)
23 \( 1 + 249. iT - 2.79e5T^{2} \)
29 \( 1 + 416. iT - 7.07e5T^{2} \)
31 \( 1 - 1.08e3iT - 9.23e5T^{2} \)
37 \( 1 + 742. iT - 1.87e6T^{2} \)
41 \( 1 - 1.19e3T + 2.82e6T^{2} \)
43 \( 1 + 2.25e3T + 3.41e6T^{2} \)
47 \( 1 + 2.79e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.87e3iT - 7.89e6T^{2} \)
59 \( 1 - 5.25e3T + 1.21e7T^{2} \)
61 \( 1 + 1.36e3iT - 1.38e7T^{2} \)
67 \( 1 + 316.T + 2.01e7T^{2} \)
71 \( 1 + 8.96e3iT - 2.54e7T^{2} \)
73 \( 1 + 9.72e3T + 2.83e7T^{2} \)
79 \( 1 - 5.46e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.06e4T + 4.74e7T^{2} \)
89 \( 1 + 6.00e3T + 6.27e7T^{2} \)
97 \( 1 - 7.30e3T + 8.85e7T^{2} \)
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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

−13.29113442297297485018934341455, −11.59908790157788077759379623224, −10.44371685875061902638459859093, −9.634171898472767290775539318614, −8.491984951975585990674856151863, −7.22111101788875447112530917327, −6.79611096446234452176820716885, −4.11007946138113761234695462528, −3.20906605148495097958932794686, −2.00918425621220278403921802539, 1.32054362175796475633973491835, 2.79119059054889174194049203806, 4.23043907873983868681884266792, 5.54556086301526975909111724235, 7.59796394089058595967417388784, 8.488627377776261837286354608424, 9.059398192696331364571492772233, 9.861374680377803837079769338096, 11.82144187173997871797289841296, 12.90857437758921897355059069722

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