Properties

Degree $2$
Conductor $128$
Sign $-0.0977 + 0.995i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.28 − 7.93i)3-s + (11.2 − 4.67i)5-s + (11.8 − 11.8i)7-s + (−33.1 − 33.1i)9-s + (23.5 + 56.9i)11-s + (−13.6 − 5.65i)13-s − 104. i·15-s + 44.1i·17-s + (−66.5 − 27.5i)19-s + (−55.0 − 133. i)21-s + (−60.0 − 60.0i)23-s + (17.1 − 17.1i)25-s + (−157. + 65.1i)27-s + (−14.3 + 34.6i)29-s + 174.·31-s + ⋯
L(s)  = 1  + (0.632 − 1.52i)3-s + (1.00 − 0.418i)5-s + (0.639 − 0.639i)7-s + (−1.22 − 1.22i)9-s + (0.646 + 1.56i)11-s + (−0.291 − 0.120i)13-s − 1.80i·15-s + 0.630i·17-s + (−0.803 − 0.332i)19-s + (−0.572 − 1.38i)21-s + (−0.544 − 0.544i)23-s + (0.137 − 0.137i)25-s + (−1.12 + 0.464i)27-s + (−0.0917 + 0.221i)29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.0977 + 0.995i$
Motivic weight: \(3\)
Character: $\chi_{128} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.59671 - 1.76130i\)
\(L(\frac12)\) \(\approx\) \(1.59671 - 1.76130i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.28 + 7.93i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (-11.2 + 4.67i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (-11.8 + 11.8i)T - 343iT^{2} \)
11 \( 1 + (-23.5 - 56.9i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (13.6 + 5.65i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 44.1iT - 4.91e3T^{2} \)
19 \( 1 + (66.5 + 27.5i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (60.0 + 60.0i)T + 1.21e4iT^{2} \)
29 \( 1 + (14.3 - 34.6i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 174.T + 2.97e4T^{2} \)
37 \( 1 + (118. - 49.0i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-15.5 - 15.5i)T + 6.89e4iT^{2} \)
43 \( 1 + (-87.3 - 210. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 228. iT - 1.03e5T^{2} \)
53 \( 1 + (-258. - 624. i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (456. - 188. i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-242. + 584. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (332. - 802. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-550. + 550. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-69.2 - 69.2i)T + 3.89e5iT^{2} \)
79 \( 1 + 518. iT - 4.93e5T^{2} \)
83 \( 1 + (-595. - 246. i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-656. + 656. i)T - 7.04e5iT^{2} \)
97 \( 1 + 388.T + 9.12e5T^{2} \)
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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

−12.71294223942928942132156098344, −12.04795120643452215669285735731, −10.40934972555618472458933310051, −9.256848631938951281096655777590, −8.151128264547069740391078951379, −7.16982336589491968354086535962, −6.24050168337498229154214663690, −4.51450588605438279374398786627, −2.21496667219671740595272903127, −1.36936912528589012568040150094, 2.38677789699654888851793860036, 3.71760633136087666030224876667, 5.16421280643056573285396486827, 6.18429938529307767703005346820, 8.278255936016456872235088961031, 9.062612092393667158613297174784, 9.927319494150176418790216820671, 10.84479206300780068875674388825, 11.80070148073092410978083677045, 13.71125554181337942254029153401

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