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Properties

Label 2-272-272.155-c2-0-37
Degree $2$
Conductor $272$
Sign $-0.990 - 0.138i$
Analytic cond. $7.41146$
Root an. cond. $2.72240$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.51 − 1.30i)2-s + (−5.10 − 2.11i)3-s + (0.590 + 3.95i)4-s + (−1.23 + 2.97i)5-s + (4.97 + 9.86i)6-s + (5.08 − 12.2i)7-s + (4.27 − 6.76i)8-s + (15.2 + 15.2i)9-s + (5.74 − 2.89i)10-s + (10.2 − 4.23i)11-s + (5.35 − 21.4i)12-s + (−9.70 − 9.70i)13-s + (−23.7 + 11.9i)14-s + (12.5 − 12.5i)15-s + (−15.3 + 4.67i)16-s + (11.9 − 12.0i)17-s + ⋯
L(s)  = 1  + (−0.757 − 0.652i)2-s + (−1.70 − 0.704i)3-s + (0.147 + 0.989i)4-s + (−0.246 + 0.594i)5-s + (0.828 + 1.64i)6-s + (0.725 − 1.75i)7-s + (0.533 − 0.845i)8-s + (1.69 + 1.69i)9-s + (0.574 − 0.289i)10-s + (0.929 − 0.385i)11-s + (0.445 − 1.78i)12-s + (−0.746 − 0.746i)13-s + (−1.69 + 0.853i)14-s + (0.838 − 0.838i)15-s + (−0.956 + 0.292i)16-s + (0.704 − 0.709i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.990 - 0.138i$
Analytic conductor: \(7.41146\)
Root analytic conductor: \(2.72240\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :1),\ -0.990 - 0.138i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0307726 + 0.441039i\)
\(L(\frac12)\) \(\approx\) \(0.0307726 + 0.441039i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.51 + 1.30i)T \)
17 \( 1 + (-11.9 + 12.0i)T \)
good3 \( 1 + (5.10 + 2.11i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (1.23 - 2.97i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (-5.08 + 12.2i)T + (-34.6 - 34.6i)T^{2} \)
11 \( 1 + (-10.2 + 4.23i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (9.70 + 9.70i)T + 169iT^{2} \)
19 \( 1 - 12.5iT - 361T^{2} \)
23 \( 1 + (-5.48 - 2.27i)T + (374. + 374. i)T^{2} \)
29 \( 1 + (7.99 + 3.31i)T + (594. + 594. i)T^{2} \)
31 \( 1 + (15.9 + 38.5i)T + (-679. + 679. i)T^{2} \)
37 \( 1 + (19.5 + 8.11i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (9.19 + 3.80i)T + (1.18e3 + 1.18e3i)T^{2} \)
43 \( 1 - 26.4iT - 1.84e3T^{2} \)
47 \( 1 + 49.0T + 2.20e3T^{2} \)
53 \( 1 + 13.9iT - 2.80e3T^{2} \)
59 \( 1 + 51.9iT - 3.48e3T^{2} \)
61 \( 1 + (-4.84 - 11.6i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (70.8 + 70.8i)T + 4.48e3iT^{2} \)
71 \( 1 + (-97.1 + 40.2i)T + (3.56e3 - 3.56e3i)T^{2} \)
73 \( 1 + (123. - 51.1i)T + (3.76e3 - 3.76e3i)T^{2} \)
79 \( 1 + (-2.95 + 7.12i)T + (-4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 - 58.4iT - 6.88e3T^{2} \)
89 \( 1 - 127.T + 7.92e3T^{2} \)
97 \( 1 + (-4.98 - 12.0i)T + (-6.65e3 + 6.65e3i)T^{2} \)
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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

−11.22284956355779028036732892569, −10.55874972302911533169977221857, −9.813190092347078371784116759294, −7.75899274836529919924528341031, −7.41615805569444862672191894015, −6.52503016849758132877892411762, −4.96833305870704736785984915198, −3.68334073973988623408117255877, −1.42262615955818342262994053030, −0.41082561453369098999143380833, 1.51720617722375481936030077116, 4.62579957036319452304502899720, 5.18097303218048499790469693292, 6.07940925469308507576076581970, 7.04458783468841322899756239226, 8.659466387866999212627977839301, 9.222438653230674324488296837567, 10.22156135624722842565544907784, 11.30159015207476715415942103950, 11.95671697127041663302470673937

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