Properties

Label 2-272-17.16-c3-0-9
Degree $2$
Conductor $272$
Sign $0.174 - 0.984i$
Analytic cond. $16.0485$
Root an. cond. $4.00606$
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  + 7.36i·3-s − 10.1i·5-s − 17.4i·7-s − 27.2·9-s + 51.5i·11-s + 75.2·13-s + 74.4·15-s + (−12.2 + 69.0i)17-s + 28·19-s + 128.·21-s + 19.1i·23-s + 22.8·25-s − 1.72i·27-s + 70.7i·29-s − 41.4i·31-s + ⋯
L(s)  = 1  + 1.41i·3-s − 0.903i·5-s − 0.943i·7-s − 1.00·9-s + 1.41i·11-s + 1.60·13-s + 1.28·15-s + (−0.174 + 0.984i)17-s + 0.338·19-s + 1.33·21-s + 0.173i·23-s + 0.182·25-s − 0.0122i·27-s + 0.452i·29-s − 0.240i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $0.174 - 0.984i$
Analytic conductor: \(16.0485\)
Root analytic conductor: \(4.00606\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :3/2),\ 0.174 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.836960741\)
\(L(\frac12)\) \(\approx\) \(1.836960741\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (12.2 - 69.0i)T \)
good3 \( 1 - 7.36iT - 27T^{2} \)
5 \( 1 + 10.1iT - 125T^{2} \)
7 \( 1 + 17.4iT - 343T^{2} \)
11 \( 1 - 51.5iT - 1.33e3T^{2} \)
13 \( 1 - 75.2T + 2.19e3T^{2} \)
19 \( 1 - 28T + 6.85e3T^{2} \)
23 \( 1 - 19.1iT - 1.21e4T^{2} \)
29 \( 1 - 70.7iT - 2.43e4T^{2} \)
31 \( 1 + 41.4iT - 2.97e4T^{2} \)
37 \( 1 - 135. iT - 5.06e4T^{2} \)
41 \( 1 - 288. iT - 6.89e4T^{2} \)
43 \( 1 + 88.2T + 7.95e4T^{2} \)
47 \( 1 + 157.T + 1.03e5T^{2} \)
53 \( 1 - 120.T + 1.48e5T^{2} \)
59 \( 1 - 696.T + 2.05e5T^{2} \)
61 \( 1 + 683. iT - 2.26e5T^{2} \)
67 \( 1 + 123.T + 3.00e5T^{2} \)
71 \( 1 + 225. iT - 3.57e5T^{2} \)
73 \( 1 - 919. iT - 3.89e5T^{2} \)
79 \( 1 - 354. iT - 4.93e5T^{2} \)
83 \( 1 - 955.T + 5.71e5T^{2} \)
89 \( 1 - 617.T + 7.04e5T^{2} \)
97 \( 1 + 428. iT - 9.12e5T^{2} \)
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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

−11.41486727848072677056344823432, −10.53892914289100836038892412715, −9.877006483350696237000692639240, −8.985242380651871381151096285061, −8.068657600064180299092164716480, −6.64697839752185359816785720403, −5.20914196008849948466424555207, −4.34436821062144865713286689474, −3.65007862120171194154545927916, −1.33535045370642008471297947863, 0.825916877904518928838301523312, 2.37065474905028486258482461663, 3.39798344538715395176162583128, 5.64719395424340608427659999785, 6.31656103451142006192781817718, 7.19272899340175634812227521762, 8.323683200036026033811553437896, 8.985288078032200279468264795833, 10.64328891779203615123515754300, 11.44178199906520629197961186319

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