Properties

Label 2-272-17.13-c3-0-8
Degree $2$
Conductor $272$
Sign $-0.847 - 0.531i$
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  + (5.11 + 5.11i)3-s + (7.20 + 7.20i)5-s + (−19.2 + 19.2i)7-s + 25.3i·9-s + (−5.30 + 5.30i)11-s − 59.4·13-s + 73.6i·15-s + (−7.18 − 69.7i)17-s + 114. i·19-s − 196.·21-s + (63.2 − 63.2i)23-s − 21.2i·25-s + (8.60 − 8.60i)27-s + (−10.6 − 10.6i)29-s + (143. + 143. i)31-s + ⋯
L(s)  = 1  + (0.984 + 0.984i)3-s + (0.644 + 0.644i)5-s + (−1.03 + 1.03i)7-s + 0.937i·9-s + (−0.145 + 0.145i)11-s − 1.26·13-s + 1.26i·15-s + (−0.102 − 0.994i)17-s + 1.38i·19-s − 2.04·21-s + (0.573 − 0.573i)23-s − 0.169i·25-s + (0.0613 − 0.0613i)27-s + (−0.0685 − 0.0685i)29-s + (0.830 + 0.830i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.847 - 0.531i$
Analytic conductor: \(16.0485\)
Root analytic conductor: \(4.00606\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :3/2),\ -0.847 - 0.531i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.021998652\)
\(L(\frac12)\) \(\approx\) \(2.021998652\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.18 + 69.7i)T \)
good3 \( 1 + (-5.11 - 5.11i)T + 27iT^{2} \)
5 \( 1 + (-7.20 - 7.20i)T + 125iT^{2} \)
7 \( 1 + (19.2 - 19.2i)T - 343iT^{2} \)
11 \( 1 + (5.30 - 5.30i)T - 1.33e3iT^{2} \)
13 \( 1 + 59.4T + 2.19e3T^{2} \)
19 \( 1 - 114. iT - 6.85e3T^{2} \)
23 \( 1 + (-63.2 + 63.2i)T - 1.21e4iT^{2} \)
29 \( 1 + (10.6 + 10.6i)T + 2.43e4iT^{2} \)
31 \( 1 + (-143. - 143. i)T + 2.97e4iT^{2} \)
37 \( 1 + (-170. - 170. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-243. + 243. i)T - 6.89e4iT^{2} \)
43 \( 1 - 201. iT - 7.95e4T^{2} \)
47 \( 1 + 422.T + 1.03e5T^{2} \)
53 \( 1 - 538. iT - 1.48e5T^{2} \)
59 \( 1 - 330. iT - 2.05e5T^{2} \)
61 \( 1 + (404. - 404. i)T - 2.26e5iT^{2} \)
67 \( 1 + 40.2T + 3.00e5T^{2} \)
71 \( 1 + (-694. - 694. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-541. - 541. i)T + 3.89e5iT^{2} \)
79 \( 1 + (585. - 585. i)T - 4.93e5iT^{2} \)
83 \( 1 + 670. iT - 5.71e5T^{2} \)
89 \( 1 + 130.T + 7.04e5T^{2} \)
97 \( 1 + (262. + 262. i)T + 9.12e5iT^{2} \)
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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

−11.97202758729498831875032940805, −10.44033985531579817253041327022, −9.767927506293868017457525746061, −9.346844806955611460829126988924, −8.249501955617383199036038135102, −6.88825517023753621135542312953, −5.80018161442678700364373545442, −4.52923209724168213634683771352, −2.95176000644226358124568264110, −2.58521235417313578577798177018, 0.65448792568703826252007015848, 2.10908033258797103348757159109, 3.29475864299757027528285741697, 4.82993521029543724102542190600, 6.36786616638300661720699423885, 7.21376730903044810711389155664, 8.043752521813030247462325422546, 9.269139881521361065905810618297, 9.760045433890892224545449981860, 11.04684686921309671421090170802

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