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: $\chi_{128} (113, \cdot )$
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

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

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