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Properties

Label 2-2e7-16.5-c3-0-5
Degree $2$
Conductor $128$
Sign $0.875 - 0.483i$
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  + (3.27 − 3.27i)3-s + (12.6 + 12.6i)5-s + 13.8i·7-s + 5.59i·9-s + (−1.54 − 1.54i)11-s + (−32.7 + 32.7i)13-s + 82.7·15-s + 18.6·17-s + (86.4 − 86.4i)19-s + (45.3 + 45.3i)21-s − 134. i·23-s + 194. i·25-s + (106. + 106. i)27-s + (59.7 − 59.7i)29-s − 31.5·31-s + ⋯
L(s)  = 1  + (0.629 − 0.629i)3-s + (1.13 + 1.13i)5-s + 0.749i·7-s + 0.207i·9-s + (−0.0424 − 0.0424i)11-s + (−0.699 + 0.699i)13-s + 1.42·15-s + 0.266·17-s + (1.04 − 1.04i)19-s + (0.471 + 0.471i)21-s − 1.21i·23-s + 1.55i·25-s + (0.760 + 0.760i)27-s + (0.382 − 0.382i)29-s − 0.182·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.875 - 0.483i$
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: \((2,\ 128,\ (\ :3/2),\ 0.875 - 0.483i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.22395 + 0.573413i\)
\(L(\frac12)\) \(\approx\) \(2.22395 + 0.573413i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.27 + 3.27i)T - 27iT^{2} \)
5 \( 1 + (-12.6 - 12.6i)T + 125iT^{2} \)
7 \( 1 - 13.8iT - 343T^{2} \)
11 \( 1 + (1.54 + 1.54i)T + 1.33e3iT^{2} \)
13 \( 1 + (32.7 - 32.7i)T - 2.19e3iT^{2} \)
17 \( 1 - 18.6T + 4.91e3T^{2} \)
19 \( 1 + (-86.4 + 86.4i)T - 6.85e3iT^{2} \)
23 \( 1 + 134. iT - 1.21e4T^{2} \)
29 \( 1 + (-59.7 + 59.7i)T - 2.43e4iT^{2} \)
31 \( 1 + 31.5T + 2.97e4T^{2} \)
37 \( 1 + (89.1 + 89.1i)T + 5.06e4iT^{2} \)
41 \( 1 + 210. iT - 6.89e4T^{2} \)
43 \( 1 + (119. + 119. i)T + 7.95e4iT^{2} \)
47 \( 1 + 182.T + 1.03e5T^{2} \)
53 \( 1 + (-26.1 - 26.1i)T + 1.48e5iT^{2} \)
59 \( 1 + (441. + 441. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-174. + 174. i)T - 2.26e5iT^{2} \)
67 \( 1 + (91.7 - 91.7i)T - 3.00e5iT^{2} \)
71 \( 1 - 348. iT - 3.57e5T^{2} \)
73 \( 1 + 299. iT - 3.89e5T^{2} \)
79 \( 1 + 943.T + 4.93e5T^{2} \)
83 \( 1 + (313. - 313. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.41e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−13.18313477752523572436105588715, −12.02785021575525922529681544991, −10.80262867707699742592933120950, −9.747898149982149951024776030061, −8.775448945161175367054712167159, −7.36621382743666265677352873715, −6.53065461169934764988787980956, −5.20157180158089171189780341503, −2.83397966567345261400116057101, −2.09006196233956034109291695961, 1.28284901175552888252646733892, 3.28775417846406035560790772638, 4.75755560745901467626731403101, 5.82567651682272757540315407525, 7.57061017635938452393979892101, 8.767074957154685838973329678838, 9.801519614779622573333715116115, 10.11459041376633003028696903419, 11.93839991496962270120926704638, 12.96522987370171854082617110137

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