Properties

Degree $2$
Conductor $128$
Sign $0.439 - 0.898i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.299 + 0.723i)3-s + (1.34 + 3.25i)5-s + (−0.583 − 0.583i)7-s + (5.93 + 5.93i)9-s + (3.03 + 7.33i)11-s + (−6.38 + 15.4i)13-s − 2.75·15-s − 19.0i·17-s + (29.6 + 12.2i)19-s + (0.596 − 0.247i)21-s + (−15.2 + 15.2i)23-s + (8.91 − 8.91i)25-s + (−12.5 + 5.21i)27-s + (−20.5 − 8.49i)29-s − 53.6i·31-s + ⋯
L(s)  = 1  + (−0.0999 + 0.241i)3-s + (0.269 + 0.650i)5-s + (−0.0833 − 0.0833i)7-s + (0.658 + 0.658i)9-s + (0.276 + 0.666i)11-s + (−0.491 + 1.18i)13-s − 0.183·15-s − 1.12i·17-s + (1.56 + 0.646i)19-s + (0.0284 − 0.0117i)21-s + (−0.665 + 0.665i)23-s + (0.356 − 0.356i)25-s + (−0.466 + 0.193i)27-s + (−0.707 − 0.293i)29-s − 1.73i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.439 - 0.898i$
Motivic weight: \(2\)
Character: $\chi_{128} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.439 - 0.898i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19649 + 0.746221i\)
\(L(\frac12)\) \(\approx\) \(1.19649 + 0.746221i\)
\(L(2)\) 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 \( 1 + (0.299 - 0.723i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (-1.34 - 3.25i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (0.583 + 0.583i)T + 49iT^{2} \)
11 \( 1 + (-3.03 - 7.33i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (6.38 - 15.4i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 19.0iT - 289T^{2} \)
19 \( 1 + (-29.6 - 12.2i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (15.2 - 15.2i)T - 529iT^{2} \)
29 \( 1 + (20.5 + 8.49i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 53.6iT - 961T^{2} \)
37 \( 1 + (3.80 + 9.17i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-14.5 - 14.5i)T + 1.68e3iT^{2} \)
43 \( 1 + (20.3 + 49.1i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 4.73T + 2.20e3T^{2} \)
53 \( 1 + (-61.4 + 25.4i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (42.4 - 17.5i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (27.7 + 11.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-9.42 + 22.7i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-95.1 - 95.1i)T + 5.04e3iT^{2} \)
73 \( 1 + (-37.1 - 37.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 70.3T + 6.24e3T^{2} \)
83 \( 1 + (14.5 + 6.01i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (60.8 - 60.8i)T - 7.92e3iT^{2} \)
97 \( 1 - 31.8T + 9.40e3T^{2} \)
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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

−13.53082887199716508898705518176, −12.07419067989537701685852185840, −11.30075137212518431030557607418, −9.898236221995572319731953435731, −9.563382707477651818220428625393, −7.62416410667233458114485024680, −6.90117822575783999522917272869, −5.33439254763392983172071379417, −4.00933243209205149591886716926, −2.13719076209485208375755958102, 1.11453346237306161341418279583, 3.35451900483681258665433294192, 5.04464951297116182429950068087, 6.19528648353782303222497981996, 7.51542153237002350933904439943, 8.736019152819743948489281896506, 9.721147387215065132503846082542, 10.82714501545066621508740285930, 12.23231825308051132237861963026, 12.74196967702268814780331456150

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