Properties

Label 2-272-17.16-c3-0-7
Degree $2$
Conductor $272$
Sign $-0.973 - 0.230i$
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  + 4.44i·3-s + 19.4i·5-s + 14.9i·7-s + 7.23·9-s + 31.1i·11-s − 5.21·13-s − 86.4·15-s + (68.2 + 16.1i)17-s + 28·19-s − 66.6·21-s − 167. i·23-s − 252.·25-s + 152. i·27-s − 136. i·29-s − 50.5i·31-s + ⋯
L(s)  = 1  + 0.855i·3-s + 1.73i·5-s + 0.809i·7-s + 0.267·9-s + 0.853i·11-s − 0.111·13-s − 1.48·15-s + (0.973 + 0.230i)17-s + 0.338·19-s − 0.692·21-s − 1.51i·23-s − 2.02·25-s + 1.08i·27-s − 0.871i·29-s − 0.292i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.973 - 0.230i$
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: \((2,\ 272,\ (\ :3/2),\ -0.973 - 0.230i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.753660746\)
\(L(\frac12)\) \(\approx\) \(1.753660746\)
\(L(\frac{5}{2})\) 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 \( 1 + (-68.2 - 16.1i)T \)
good3 \( 1 - 4.44iT - 27T^{2} \)
5 \( 1 - 19.4iT - 125T^{2} \)
7 \( 1 - 14.9iT - 343T^{2} \)
11 \( 1 - 31.1iT - 1.33e3T^{2} \)
13 \( 1 + 5.21T + 2.19e3T^{2} \)
19 \( 1 - 28T + 6.85e3T^{2} \)
23 \( 1 + 167. iT - 1.21e4T^{2} \)
29 \( 1 + 136. iT - 2.43e4T^{2} \)
31 \( 1 + 50.5iT - 2.97e4T^{2} \)
37 \( 1 - 260. iT - 5.06e4T^{2} \)
41 \( 1 + 183. iT - 6.89e4T^{2} \)
43 \( 1 - 348.T + 7.95e4T^{2} \)
47 \( 1 + 318.T + 1.03e5T^{2} \)
53 \( 1 + 408.T + 1.48e5T^{2} \)
59 \( 1 + 108.T + 2.05e5T^{2} \)
61 \( 1 - 123. iT - 2.26e5T^{2} \)
67 \( 1 - 243.T + 3.00e5T^{2} \)
71 \( 1 - 42.7iT - 3.57e5T^{2} \)
73 \( 1 + 875. iT - 3.89e5T^{2} \)
79 \( 1 - 750. iT - 4.93e5T^{2} \)
83 \( 1 - 472.T + 5.71e5T^{2} \)
89 \( 1 - 376.T + 7.04e5T^{2} \)
97 \( 1 - 303. iT - 9.12e5T^{2} \)
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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

−11.76470770780111847693815459398, −10.71061592417919901735505102267, −10.11432296767543238494447938713, −9.415753166149279927856834863908, −7.931976520781335138981494286659, −6.93670366358621845677797323506, −5.96085506865641306019120225892, −4.59960934722369966216956883390, −3.37162438896962760982434464865, −2.29668252585318136261265721065, 0.74055304948636410348533807081, 1.46126581980983879775679778368, 3.63859512478006888677454889118, 4.91086513169062701552069607786, 5.88806023522228958601623205952, 7.35689109916127315112995066043, 7.950475316337965870719207906950, 9.042981702409286903501396146822, 9.903652159069573080842600287662, 11.26016897375270437546936502729

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