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Properties

Label 2-2e7-16.13-c3-0-8
Degree $2$
Conductor $128$
Sign $-0.483 + 0.875i$
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

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

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.483 + 0.875i$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.483 + 0.875i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.670154 - 1.13579i\)
\(L(\frac12)\) \(\approx\) \(0.670154 - 1.13579i\)
\(L(\frac{5}{2})\) 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

−12.67188366051350463558344247175, −11.85384250301024020452756145407, −10.40022648199925015166587646180, −9.263315272965089253733276187560, −8.448200868229780336654604260242, −6.73408721364126725352949659270, −5.76946461758849453315252436969, −4.86978075734467115733116009140, −2.33616594402393225261548386729, −0.72746705590089028039057230126, 2.13572833002157038585733130948, 3.97749720127970560908949189925, 5.43766164483573018685714363875, 6.47574726169490169260452060889, 7.58804450765820190788745155088, 9.467403888782344626711385241688, 10.31770807799437992602459024464, 10.76245701456764122678890190076, 11.97758683516929906352774586758, 13.48845681870954573521813068359

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