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Properties

Label 2-2e7-16.5-c3-0-4
Degree $2$
Conductor $128$
Sign $0.851 + 0.524i$
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  + (−5.96 + 5.96i)3-s + (−8.67 − 8.67i)5-s − 1.63i·7-s − 44.1i·9-s + (18.2 + 18.2i)11-s + (9.34 − 9.34i)13-s + 103.·15-s + 53.6·17-s + (70.9 − 70.9i)19-s + (9.77 + 9.77i)21-s + 25.1i·23-s + 25.6i·25-s + (102. + 102. i)27-s + (181. − 181. i)29-s − 132.·31-s + ⋯
L(s)  = 1  + (−1.14 + 1.14i)3-s + (−0.776 − 0.776i)5-s − 0.0885i·7-s − 1.63i·9-s + (0.498 + 0.498i)11-s + (0.199 − 0.199i)13-s + 1.78·15-s + 0.764·17-s + (0.857 − 0.857i)19-s + (0.101 + 0.101i)21-s + 0.227i·23-s + 0.205i·25-s + (0.729 + 0.729i)27-s + (1.15 − 1.15i)29-s − 0.768·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.851 + 0.524i$
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: \((2,\ 128,\ (\ :3/2),\ 0.851 + 0.524i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.770584 - 0.218146i\)
\(L(\frac12)\) \(\approx\) \(0.770584 - 0.218146i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.96 - 5.96i)T - 27iT^{2} \)
5 \( 1 + (8.67 + 8.67i)T + 125iT^{2} \)
7 \( 1 + 1.63iT - 343T^{2} \)
11 \( 1 + (-18.2 - 18.2i)T + 1.33e3iT^{2} \)
13 \( 1 + (-9.34 + 9.34i)T - 2.19e3iT^{2} \)
17 \( 1 - 53.6T + 4.91e3T^{2} \)
19 \( 1 + (-70.9 + 70.9i)T - 6.85e3iT^{2} \)
23 \( 1 - 25.1iT - 1.21e4T^{2} \)
29 \( 1 + (-181. + 181. i)T - 2.43e4iT^{2} \)
31 \( 1 + 132.T + 2.97e4T^{2} \)
37 \( 1 + (174. + 174. i)T + 5.06e4iT^{2} \)
41 \( 1 + 198. iT - 6.89e4T^{2} \)
43 \( 1 + (285. + 285. i)T + 7.95e4iT^{2} \)
47 \( 1 + 78.3T + 1.03e5T^{2} \)
53 \( 1 + (-525. - 525. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-46.5 - 46.5i)T + 2.05e5iT^{2} \)
61 \( 1 + (193. - 193. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-282. + 282. i)T - 3.00e5iT^{2} \)
71 \( 1 + 727. iT - 3.57e5T^{2} \)
73 \( 1 + 106. iT - 3.89e5T^{2} \)
79 \( 1 - 58.9T + 4.93e5T^{2} \)
83 \( 1 + (410. - 410. i)T - 5.71e5iT^{2} \)
89 \( 1 + 768. iT - 7.04e5T^{2} \)
97 \( 1 + 809.T + 9.12e5T^{2} \)
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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

−12.20334367727350597180143166025, −11.87873157367222981483598676796, −10.74317258972481362575039040809, −9.796628209246461933198981530520, −8.758192041383591517293502881443, −7.26770980424429554317165832486, −5.70662989083466118283831572522, −4.73752138700451558273124553279, −3.78355208094829524610263129481, −0.57898593630610030374669476513, 1.21631403091551861322041598964, 3.40063727690054697750028605202, 5.35112375078951400164739979315, 6.50537861571995102399863130692, 7.26008374636979477813477689161, 8.339630692470800164302142824061, 10.18441483792375940948731451563, 11.32096216793748018545693473354, 11.80026405268790416113807584859, 12.66210105410185820619412306976

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