Properties

Degree $2$
Conductor $128$
Sign $-0.167 + 0.985i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.53 − 2.29i)3-s + (4.22 + 10.1i)5-s + (11.6 − 11.6i)7-s + (6.27 + 6.27i)9-s + (−27.7 + 11.4i)11-s + (15.9 − 38.5i)13-s − 66.0i·15-s − 131. i·17-s + (13.7 − 33.3i)19-s + (−91.1 + 37.7i)21-s + (−128. − 128. i)23-s + (2.32 − 2.32i)25-s + (41.5 + 100. i)27-s + (−192. − 79.5i)29-s + 215.·31-s + ⋯
L(s)  = 1  + (−1.06 − 0.441i)3-s + (0.377 + 0.911i)5-s + (0.628 − 0.628i)7-s + (0.232 + 0.232i)9-s + (−0.760 + 0.315i)11-s + (0.340 − 0.822i)13-s − 1.13i·15-s − 1.87i·17-s + (0.166 − 0.402i)19-s + (−0.946 + 0.392i)21-s + (−1.16 − 1.16i)23-s + (0.0186 − 0.0186i)25-s + (0.296 + 0.715i)27-s + (−1.23 − 0.509i)29-s + 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.167 + 0.985i$
Motivic weight: \(3\)
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.167 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.591173 - 0.700356i\)
\(L(\frac12)\) \(\approx\) \(0.591173 - 0.700356i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.53 + 2.29i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-4.22 - 10.1i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-11.6 + 11.6i)T - 343iT^{2} \)
11 \( 1 + (27.7 - 11.4i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-15.9 + 38.5i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 131. iT - 4.91e3T^{2} \)
19 \( 1 + (-13.7 + 33.3i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (128. + 128. i)T + 1.21e4iT^{2} \)
29 \( 1 + (192. + 79.5i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 215.T + 2.97e4T^{2} \)
37 \( 1 + (9.39 + 22.6i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-257. - 257. i)T + 6.89e4iT^{2} \)
43 \( 1 + (-81.5 + 33.7i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 113. iT - 1.03e5T^{2} \)
53 \( 1 + (11.8 - 4.92i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (210. + 508. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (251. + 104. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-418. - 173. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-28.0 + 28.0i)T - 3.57e5iT^{2} \)
73 \( 1 + (333. + 333. i)T + 3.89e5iT^{2} \)
79 \( 1 + 38.8iT - 4.93e5T^{2} \)
83 \( 1 + (229. - 555. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (872. - 872. i)T - 7.04e5iT^{2} \)
97 \( 1 + 51.1T + 9.12e5T^{2} \)
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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

−12.47344288477896870304681986224, −11.37023584722079189772377475264, −10.76857231081333766197550824983, −9.791009842376639562987137885833, −7.932493414061536198958458305467, −6.99313955239718230807618917789, −5.94957146614376145906952308760, −4.77438973243137929008210179051, −2.67930694730413466394691473158, −0.54614934153372723013186166382, 1.68118793375683991664398178379, 4.21151586730851632535892248060, 5.47041941520643156883188407396, 5.94900626234415969934395938068, 7.988103034005195489394555064451, 8.934395174259017260423314278061, 10.20770171186552631122480569439, 11.15609988455098667620575112774, 12.01501577639345622779109764826, 12.94996912641603773865114705567

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