Properties

Degree $2$
Conductor $128$
Sign $0.675 + 0.737i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.998 − 2.40i)3-s + (17.4 − 7.20i)5-s + (−4.37 + 4.37i)7-s + (14.2 + 14.2i)9-s + (−11.7 − 28.4i)11-s + (−12.9 − 5.35i)13-s − 49.1i·15-s − 72.9i·17-s + (143. + 59.2i)19-s + (6.17 + 14.8i)21-s + (−83.6 − 83.6i)23-s + (162. − 162. i)25-s + (113. − 47.1i)27-s + (−39.6 + 95.7i)29-s + 29.0·31-s + ⋯
L(s)  = 1  + (0.192 − 0.463i)3-s + (1.55 − 0.644i)5-s + (−0.236 + 0.236i)7-s + (0.528 + 0.528i)9-s + (−0.323 − 0.780i)11-s + (−0.275 − 0.114i)13-s − 0.845i·15-s − 1.04i·17-s + (1.72 + 0.715i)19-s + (0.0641 + 0.154i)21-s + (−0.758 − 0.758i)23-s + (1.29 − 1.29i)25-s + (0.810 − 0.335i)27-s + (−0.253 + 0.613i)29-s + 0.168·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.675 + 0.737i$
Motivic weight: \(3\)
Character: $\chi_{128} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.675 + 0.737i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.99772 - 0.879725i\)
\(L(\frac12)\) \(\approx\) \(1.99772 - 0.879725i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.998 + 2.40i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (-17.4 + 7.20i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (4.37 - 4.37i)T - 343iT^{2} \)
11 \( 1 + (11.7 + 28.4i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (12.9 + 5.35i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 72.9iT - 4.91e3T^{2} \)
19 \( 1 + (-143. - 59.2i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (83.6 + 83.6i)T + 1.21e4iT^{2} \)
29 \( 1 + (39.6 - 95.7i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 29.0T + 2.97e4T^{2} \)
37 \( 1 + (267. - 110. i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (124. + 124. i)T + 6.89e4iT^{2} \)
43 \( 1 + (-27.0 - 65.2i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 282. iT - 1.03e5T^{2} \)
53 \( 1 + (51.4 + 124. i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-222. + 92.1i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (226. - 547. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (356. - 859. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (690. - 690. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-223. - 223. i)T + 3.89e5iT^{2} \)
79 \( 1 + 698. iT - 4.93e5T^{2} \)
83 \( 1 + (-915. - 379. i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (163. - 163. i)T - 7.04e5iT^{2} \)
97 \( 1 + 839.T + 9.12e5T^{2} \)
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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

−12.92206341528269138893001020745, −11.97899662381187415100463847965, −10.36767844778448948960029784043, −9.634668885951619361751214339397, −8.562883090324816137751528458073, −7.28023610396681270620126597401, −5.89292821898204168035043678294, −5.03816715715107914194513007653, −2.73452455669047512065575638488, −1.32312149845876963037472538092, 1.86651357459751885213767573834, 3.44861504854309206084435638701, 5.13535607156500420632834110951, 6.37330277200243915226298036335, 7.39677030631175181533115981486, 9.264086895502127489474425382649, 9.878131351248018507700562072446, 10.50026110964960562906476237876, 12.04778713369724710840611670798, 13.25204033778718558915589232986

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