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Properties

Label 2-2e7-32.5-c3-0-9
Degree $2$
Conductor $128$
Sign $-0.938 + 0.344i$
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  + (−1.94 − 4.68i)3-s + (4.93 + 2.04i)5-s + (−14.0 − 14.0i)7-s + (0.893 − 0.893i)9-s + (3.78 − 9.14i)11-s + (−64.7 + 26.8i)13-s − 27.0i·15-s − 79.3i·17-s + (−94.7 + 39.2i)19-s + (−38.6 + 93.2i)21-s + (−71.6 + 71.6i)23-s + (−68.2 − 68.2i)25-s + (−132. − 54.8i)27-s + (−53.0 − 128. i)29-s + 267.·31-s + ⋯
L(s)  = 1  + (−0.373 − 0.901i)3-s + (0.441 + 0.182i)5-s + (−0.760 − 0.760i)7-s + (0.0331 − 0.0331i)9-s + (0.103 − 0.250i)11-s + (−1.38 + 0.572i)13-s − 0.466i·15-s − 1.13i·17-s + (−1.14 + 0.474i)19-s + (−0.401 + 0.969i)21-s + (−0.649 + 0.649i)23-s + (−0.545 − 0.545i)25-s + (−0.944 − 0.391i)27-s + (−0.339 − 0.819i)29-s + 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.938 + 0.344i$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.938 + 0.344i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.143320 - 0.805385i\)
\(L(\frac12)\) \(\approx\) \(0.143320 - 0.805385i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.94 + 4.68i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (-4.93 - 2.04i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (14.0 + 14.0i)T + 343iT^{2} \)
11 \( 1 + (-3.78 + 9.14i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (64.7 - 26.8i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 79.3iT - 4.91e3T^{2} \)
19 \( 1 + (94.7 - 39.2i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (71.6 - 71.6i)T - 1.21e4iT^{2} \)
29 \( 1 + (53.0 + 128. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 267.T + 2.97e4T^{2} \)
37 \( 1 + (-205. - 85.1i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-210. + 210. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-56.9 + 137. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 173. iT - 1.03e5T^{2} \)
53 \( 1 + (188. - 455. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-627. - 260. i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-66.5 - 160. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (211. + 511. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-226. - 226. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-802. + 802. i)T - 3.89e5iT^{2} \)
79 \( 1 + 552. iT - 4.93e5T^{2} \)
83 \( 1 + (-137. + 57.1i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (579. + 579. i)T + 7.04e5iT^{2} \)
97 \( 1 + 912.T + 9.12e5T^{2} \)
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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

−12.37948283715086087879502180293, −11.67952655083132156364415027026, −10.16642170193463912830529754096, −9.518547348534476382506447901825, −7.73733340108468239353396116588, −6.84722300880296874178195228203, −6.01897279751628192796308678201, −4.22089636391652406450935073754, −2.30270153549677088124505325225, −0.41484794021145959798859763051, 2.43570696356122229262049824893, 4.25118106108360972996275576428, 5.39803693629547860501366398375, 6.48079972055866824481122305228, 8.138648964187544427362983612888, 9.529770329532709991904973852352, 9.999751040376629795299337415933, 11.08693956208639528251260120840, 12.51325846820456405478002776954, 12.96771069942841847927575541369

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