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Properties

Label 2-272-272.155-c2-0-38
Degree $2$
Conductor $272$
Sign $0.995 - 0.0896i$
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  + (−1.95 + 0.419i)2-s + (4.50 + 1.86i)3-s + (3.64 − 1.63i)4-s + (1.59 − 3.85i)5-s + (−9.59 − 1.76i)6-s + (1.19 − 2.89i)7-s + (−6.44 + 4.73i)8-s + (10.4 + 10.4i)9-s + (−1.50 + 8.21i)10-s + (7.41 − 3.06i)11-s + (19.5 − 0.578i)12-s + (−2.94 − 2.94i)13-s + (−1.12 + 6.15i)14-s + (14.4 − 14.4i)15-s + (10.6 − 11.9i)16-s + (15.1 − 7.69i)17-s + ⋯
L(s)  = 1  + (−0.977 + 0.209i)2-s + (1.50 + 0.622i)3-s + (0.912 − 0.409i)4-s + (0.319 − 0.771i)5-s + (−1.59 − 0.293i)6-s + (0.171 − 0.412i)7-s + (−0.805 + 0.591i)8-s + (1.16 + 1.16i)9-s + (−0.150 + 0.821i)10-s + (0.673 − 0.279i)11-s + (1.62 − 0.0482i)12-s + (−0.226 − 0.226i)13-s + (−0.0806 + 0.439i)14-s + (0.960 − 0.960i)15-s + (0.663 − 0.747i)16-s + (0.891 − 0.452i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $0.995 - 0.0896i$
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: \((2,\ 272,\ (\ :1),\ 0.995 - 0.0896i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.90183 + 0.0854027i\)
\(L(\frac12)\) \(\approx\) \(1.90183 + 0.0854027i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.95 - 0.419i)T \)
17 \( 1 + (-15.1 + 7.69i)T \)
good3 \( 1 + (-4.50 - 1.86i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-1.59 + 3.85i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (-1.19 + 2.89i)T + (-34.6 - 34.6i)T^{2} \)
11 \( 1 + (-7.41 + 3.06i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (2.94 + 2.94i)T + 169iT^{2} \)
19 \( 1 - 8.13iT - 361T^{2} \)
23 \( 1 + (27.9 + 11.5i)T + (374. + 374. i)T^{2} \)
29 \( 1 + (0.808 + 0.334i)T + (594. + 594. i)T^{2} \)
31 \( 1 + (-0.138 - 0.334i)T + (-679. + 679. i)T^{2} \)
37 \( 1 + (-25.2 - 10.4i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (14.4 + 5.97i)T + (1.18e3 + 1.18e3i)T^{2} \)
43 \( 1 - 34.7iT - 1.84e3T^{2} \)
47 \( 1 + 6.51T + 2.20e3T^{2} \)
53 \( 1 + 45.8iT - 2.80e3T^{2} \)
59 \( 1 - 17.1iT - 3.48e3T^{2} \)
61 \( 1 + (-35.1 - 84.7i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-13.1 - 13.1i)T + 4.48e3iT^{2} \)
71 \( 1 + (57.4 - 23.7i)T + (3.56e3 - 3.56e3i)T^{2} \)
73 \( 1 + (115. - 47.9i)T + (3.76e3 - 3.76e3i)T^{2} \)
79 \( 1 + (45.4 - 109. i)T + (-4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 - 50.1iT - 6.88e3T^{2} \)
89 \( 1 + 105.T + 7.92e3T^{2} \)
97 \( 1 + (68.1 + 164. i)T + (-6.65e3 + 6.65e3i)T^{2} \)
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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

−11.48916694821771872881409090632, −10.09547124780497395511496403100, −9.751455868935975387819257727086, −8.749723476746659779657996824069, −8.211158115458469526769637123560, −7.25175380032213553299538320078, −5.70010418280145373405666483810, −4.20460537234750886569531833353, −2.84148209940489771884192997733, −1.35861851784048328918200649956, 1.67218691960604542547917024845, 2.58906707942255608370904584855, 3.67427692945506293456474214360, 6.16863197455017470832591713053, 7.14940225722642252328362085012, 7.900949077443612440085776752997, 8.802219629348055398292720676208, 9.575581267329005544441424186246, 10.38706603336173040962477222238, 11.71530425907440201201247967766

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