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Properties

Label 2-2e7-16.5-c3-0-8
Degree $2$
Conductor $128$
Sign $-0.973 + 0.226i$
Analytic cond. $7.55224$
Root an. cond. $2.74813$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.756 − 0.756i)3-s + (−8.22 − 8.22i)5-s − 2.67i·7-s + 25.8i·9-s + (−45.2 − 45.2i)11-s + (−35.3 + 35.3i)13-s − 12.4·15-s − 72.4·17-s + (19.4 − 19.4i)19-s + (−2.02 − 2.02i)21-s − 139. i·23-s + 10.3i·25-s + (39.9 + 39.9i)27-s + (−66.0 + 66.0i)29-s − 188.·31-s + ⋯
L(s)  = 1  + (0.145 − 0.145i)3-s + (−0.735 − 0.735i)5-s − 0.144i·7-s + 0.957i·9-s + (−1.23 − 1.23i)11-s + (−0.755 + 0.755i)13-s − 0.214·15-s − 1.03·17-s + (0.234 − 0.234i)19-s + (−0.0210 − 0.0210i)21-s − 1.26i·23-s + 0.0826i·25-s + (0.285 + 0.285i)27-s + (−0.422 + 0.422i)29-s − 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.226i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.973 + 0.226i$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.973 + 0.226i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0487301 - 0.423754i\)
\(L(\frac12)\) \(\approx\) \(0.0487301 - 0.423754i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.756 + 0.756i)T - 27iT^{2} \)
5 \( 1 + (8.22 + 8.22i)T + 125iT^{2} \)
7 \( 1 + 2.67iT - 343T^{2} \)
11 \( 1 + (45.2 + 45.2i)T + 1.33e3iT^{2} \)
13 \( 1 + (35.3 - 35.3i)T - 2.19e3iT^{2} \)
17 \( 1 + 72.4T + 4.91e3T^{2} \)
19 \( 1 + (-19.4 + 19.4i)T - 6.85e3iT^{2} \)
23 \( 1 + 139. iT - 1.21e4T^{2} \)
29 \( 1 + (66.0 - 66.0i)T - 2.43e4iT^{2} \)
31 \( 1 + 188.T + 2.97e4T^{2} \)
37 \( 1 + (-84.0 - 84.0i)T + 5.06e4iT^{2} \)
41 \( 1 + 104. iT - 6.89e4T^{2} \)
43 \( 1 + (31.4 + 31.4i)T + 7.95e4iT^{2} \)
47 \( 1 - 488.T + 1.03e5T^{2} \)
53 \( 1 + (149. + 149. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-284. - 284. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-228. + 228. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-139. + 139. i)T - 3.00e5iT^{2} \)
71 \( 1 + 453. iT - 3.57e5T^{2} \)
73 \( 1 - 259. iT - 3.89e5T^{2} \)
79 \( 1 + 323.T + 4.93e5T^{2} \)
83 \( 1 + (563. - 563. i)T - 5.71e5iT^{2} \)
89 \( 1 + 866. iT - 7.04e5T^{2} \)
97 \( 1 + 936.T + 9.12e5T^{2} \)
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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

−12.52557396861788923491531625709, −11.29372681905868675571576648183, −10.53739808471319104377496886269, −8.908607693648324955627992723705, −8.170947221108921189667381993949, −7.13079273697180309411543776088, −5.37906182537608873431382203400, −4.32891957454080549906193635195, −2.47670928151610084877061776029, −0.20163518818049323043374773007, 2.58312204320429630935646757056, 3.94281970347378143155949022699, 5.44705066114471587000393800643, 7.10486034231696241376949611967, 7.72534737564448214716709650415, 9.280641541110581441191334748928, 10.22541608651181725071865569902, 11.28754788965017974045406740374, 12.32809510812336779233916695471, 13.17362659885731773536062748209

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