Properties

Degree $2$
Conductor $128$
Sign $-0.978 + 0.205i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.65 − 0.686i)3-s + (−4.13 + 9.99i)5-s + (−24.2 − 24.2i)7-s + (−16.8 + 16.8i)9-s + (−3.73 − 1.54i)11-s + (−23.2 − 56.0i)13-s + 19.4i·15-s + 26.9i·17-s + (−22.7 − 54.8i)19-s + (−56.7 − 23.5i)21-s + (−76.0 + 76.0i)23-s + (5.64 + 5.64i)25-s + (−34.8 + 84.1i)27-s + (−108. + 44.7i)29-s − 175.·31-s + ⋯
L(s)  = 1  + (0.318 − 0.132i)3-s + (−0.370 + 0.893i)5-s + (−1.30 − 1.30i)7-s + (−0.622 + 0.622i)9-s + (−0.102 − 0.0424i)11-s + (−0.495 − 1.19i)13-s + 0.333i·15-s + 0.385i·17-s + (−0.274 − 0.662i)19-s + (−0.589 − 0.244i)21-s + (−0.689 + 0.689i)23-s + (0.0451 + 0.0451i)25-s + (−0.248 + 0.599i)27-s + (−0.692 + 0.286i)29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.978 + 0.205i$
Motivic weight: \(3\)
Character: $\chi_{128} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.978 + 0.205i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0163875 - 0.157738i\)
\(L(\frac12)\) \(\approx\) \(0.0163875 - 0.157738i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.65 + 0.686i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (4.13 - 9.99i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (24.2 + 24.2i)T + 343iT^{2} \)
11 \( 1 + (3.73 + 1.54i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (23.2 + 56.0i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 26.9iT - 4.91e3T^{2} \)
19 \( 1 + (22.7 + 54.8i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (76.0 - 76.0i)T - 1.21e4iT^{2} \)
29 \( 1 + (108. - 44.7i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 175.T + 2.97e4T^{2} \)
37 \( 1 + (-129. + 311. i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-70.4 + 70.4i)T - 6.89e4iT^{2} \)
43 \( 1 + (-103. - 42.7i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 249. iT - 1.03e5T^{2} \)
53 \( 1 + (-597. - 247. i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-75.6 + 182. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (309. - 128. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-297. + 123. i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (675. + 675. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-350. + 350. i)T - 3.89e5iT^{2} \)
79 \( 1 - 564. iT - 4.93e5T^{2} \)
83 \( 1 + (105. + 254. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (448. + 448. i)T + 7.04e5iT^{2} \)
97 \( 1 + 1.76e3T + 9.12e5T^{2} \)
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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

−12.67696033878412205844752337059, −10.97866305535781347059311937654, −10.53517485813558481799656940529, −9.341048150260032681614459833517, −7.71185433826574316262226616596, −7.17124418054817221363661786143, −5.76829752774262847817379634337, −3.79871821292128143097065896576, −2.80919645765810055990259504395, −0.07263800893732150677081471978, 2.50856317796825980803436628804, 3.99044506580101991893468664718, 5.57002606119244470493650151734, 6.64695334686992953116614787699, 8.389216761430514481364906933540, 9.111099858577144299234687293171, 9.819409264606918377338636891135, 11.75755911669160206824555260424, 12.21381950479472978211912490202, 13.13284000804612994180890190208

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