Properties

Degree $2$
Conductor $128$
Sign $-0.130 + 0.991i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.936 + 2.26i)3-s + (−3.18 − 7.68i)5-s + (−3.67 − 3.67i)7-s + (2.12 + 2.12i)9-s + (−6.10 − 14.7i)11-s + (2.82 − 6.80i)13-s + 20.3·15-s − 3.67i·17-s + (1.65 + 0.686i)19-s + (11.7 − 4.86i)21-s + (8.31 − 8.31i)23-s + (−31.2 + 31.2i)25-s + (−27.1 + 11.2i)27-s + (−38.8 − 16.0i)29-s − 4.11i·31-s + ⋯
L(s)  = 1  + (−0.312 + 0.753i)3-s + (−0.636 − 1.53i)5-s + (−0.524 − 0.524i)7-s + (0.236 + 0.236i)9-s + (−0.554 − 1.33i)11-s + (0.216 − 0.523i)13-s + 1.35·15-s − 0.215i·17-s + (0.0872 + 0.0361i)19-s + (0.559 − 0.231i)21-s + (0.361 − 0.361i)23-s + (−1.24 + 1.24i)25-s + (−1.00 + 0.416i)27-s + (−1.33 − 0.555i)29-s − 0.132i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.130 + 0.991i$
Motivic weight: \(2\)
Character: $\chi_{128} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ -0.130 + 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.534081 - 0.608767i\)
\(L(\frac12)\) \(\approx\) \(0.534081 - 0.608767i\)
\(L(2)\) 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 \( 1 + (0.936 - 2.26i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (3.18 + 7.68i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (3.67 + 3.67i)T + 49iT^{2} \)
11 \( 1 + (6.10 + 14.7i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-2.82 + 6.80i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 3.67iT - 289T^{2} \)
19 \( 1 + (-1.65 - 0.686i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-8.31 + 8.31i)T - 529iT^{2} \)
29 \( 1 + (38.8 + 16.0i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 4.11iT - 961T^{2} \)
37 \( 1 + (-19.8 - 47.9i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-21.1 - 21.1i)T + 1.68e3iT^{2} \)
43 \( 1 + (-0.102 - 0.247i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 39.3T + 2.20e3T^{2} \)
53 \( 1 + (-22.6 + 9.36i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-101. + 41.9i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (14.0 + 5.81i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (3.67 - 8.87i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (75.7 + 75.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (29.0 + 29.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 2.76T + 6.24e3T^{2} \)
83 \( 1 + (-79.1 - 32.8i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-72.4 + 72.4i)T - 7.92e3iT^{2} \)
97 \( 1 - 66.0T + 9.40e3T^{2} \)
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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

−13.09637564102974062138878008904, −11.69981668546648404302331817230, −10.77760133365272386524180857170, −9.700494209615092022349563531491, −8.604368239719828141208925492759, −7.66726978606547301284297057223, −5.75270689506012497596830194380, −4.74522439944451297387640698138, −3.61971622026098801614265598168, −0.57616260069519140758037916225, 2.32426672031508758389409091186, 3.87413823580533952261358234769, 5.91244934564034591581066367382, 7.10432658220469605251706402854, 7.41216851969161827547863808507, 9.294022960885786102593425458793, 10.42682435192564276607128478765, 11.44429026215590273821651067430, 12.34514978037988830424592498389, 13.14344759790585518150494956957

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