Properties

Degree $2$
Conductor $128$
Sign $-0.512 - 0.858i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.54 − 8.56i)3-s + (−7.55 − 3.12i)5-s + (−7.16 − 7.16i)7-s + (−41.6 + 41.6i)9-s + (−0.758 + 1.83i)11-s + (71.0 − 29.4i)13-s + 75.7i·15-s + 98.5i·17-s + (−89.5 + 37.1i)19-s + (−35.9 + 86.7i)21-s + (−24.9 + 24.9i)23-s + (−41.1 − 41.1i)25-s + (273. + 113. i)27-s + (−57.8 − 139. i)29-s − 58.0·31-s + ⋯
L(s)  = 1  + (−0.682 − 1.64i)3-s + (−0.675 − 0.279i)5-s + (−0.386 − 0.386i)7-s + (−1.54 + 1.54i)9-s + (−0.0207 + 0.0501i)11-s + (1.51 − 0.628i)13-s + 1.30i·15-s + 1.40i·17-s + (−1.08 + 0.448i)19-s + (−0.373 + 0.901i)21-s + (−0.226 + 0.226i)23-s + (−0.329 − 0.329i)25-s + (1.95 + 0.808i)27-s + (−0.370 − 0.894i)29-s − 0.336·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.512 - 0.858i$
Motivic weight: \(3\)
Character: $\chi_{128} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.512 - 0.858i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.143938 + 0.253535i\)
\(L(\frac12)\) \(\approx\) \(0.143938 + 0.253535i\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.54 + 8.56i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (7.55 + 3.12i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (7.16 + 7.16i)T + 343iT^{2} \)
11 \( 1 + (0.758 - 1.83i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-71.0 + 29.4i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 98.5iT - 4.91e3T^{2} \)
19 \( 1 + (89.5 - 37.1i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (24.9 - 24.9i)T - 1.21e4iT^{2} \)
29 \( 1 + (57.8 + 139. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 58.0T + 2.97e4T^{2} \)
37 \( 1 + (202. + 84.0i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (45.3 - 45.3i)T - 6.89e4iT^{2} \)
43 \( 1 + (89.7 - 216. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 4.38iT - 1.03e5T^{2} \)
53 \( 1 + (-8.98 + 21.6i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (287. + 119. i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (28.2 + 68.0i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (293. + 708. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-579. - 579. i)T + 3.57e5iT^{2} \)
73 \( 1 + (258. - 258. i)T - 3.89e5iT^{2} \)
79 \( 1 + 834. iT - 4.93e5T^{2} \)
83 \( 1 + (234. - 97.3i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-179. - 179. i)T + 7.04e5iT^{2} \)
97 \( 1 + 624.T + 9.12e5T^{2} \)
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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

−12.41046431326553561764970818731, −11.34355548912746093267153136197, −10.49890678761307191803501646003, −8.414503709061757770740079313957, −7.87685757360368679541619326827, −6.53915379751039414352778333383, −5.84279383884856085925801538932, −3.83059231986260536631471455405, −1.63770384214424263103214222563, −0.16440523152190428156906800260, 3.32716371986607953148222005198, 4.31139230963495386695074091635, 5.56162138198210839969748968059, 6.76670800284071294609743583056, 8.676112647452904648540644803914, 9.397773140293870883974261874150, 10.67291680716501767803695972861, 11.23770412333349506515787360540, 12.10913833746893316716794342460, 13.67789588903524147618346886018

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