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Properties

Label 2-273-91.47-c1-0-8
Degree $2$
Conductor $273$
Sign $0.953 + 0.300i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.170 + 0.0455i)2-s + (0.866 − 0.5i)3-s + (−1.70 + 0.984i)4-s + (−0.0851 − 0.317i)5-s + (−0.124 + 0.124i)6-s + (2.06 − 1.64i)7-s + (0.494 − 0.494i)8-s + (0.499 − 0.866i)9-s + (0.0289 + 0.0501i)10-s + (3.55 + 0.952i)11-s + (−0.984 + 1.70i)12-s + (1.90 − 3.06i)13-s + (−0.276 + 0.374i)14-s + (−0.232 − 0.232i)15-s + (1.90 − 3.30i)16-s + (2.19 + 3.79i)17-s + ⋯
L(s)  = 1  + (−0.120 + 0.0322i)2-s + (0.499 − 0.288i)3-s + (−0.852 + 0.492i)4-s + (−0.0380 − 0.142i)5-s + (−0.0508 + 0.0508i)6-s + (0.782 − 0.623i)7-s + (0.174 − 0.174i)8-s + (0.166 − 0.288i)9-s + (0.00916 + 0.0158i)10-s + (1.07 + 0.287i)11-s + (−0.284 + 0.492i)12-s + (0.528 − 0.848i)13-s + (−0.0739 + 0.100i)14-s + (−0.0600 − 0.0600i)15-s + (0.476 − 0.825i)16-s + (0.531 + 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.953 + 0.300i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.953 + 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31295 - 0.201929i\)
\(L(\frac12)\) \(\approx\) \(1.31295 - 0.201929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-2.06 + 1.64i)T \)
13 \( 1 + (-1.90 + 3.06i)T \)
good2 \( 1 + (0.170 - 0.0455i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.0851 + 0.317i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.55 - 0.952i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.19 - 3.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.839 - 3.13i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.57 + 1.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.47T + 29T^{2} \)
31 \( 1 + (2.01 + 0.540i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.614 + 2.29i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.39 - 1.39i)T - 41iT^{2} \)
43 \( 1 - 0.148iT - 43T^{2} \)
47 \( 1 + (4.83 - 1.29i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.51 - 9.55i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.60 - 9.71i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.35 + 1.93i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.60 + 5.99i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.84 + 5.84i)T + 71iT^{2} \)
73 \( 1 + (1.31 - 4.90i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.57 - 2.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.02 - 4.02i)T - 83iT^{2} \)
89 \( 1 + (10.9 - 2.92i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (11.4 - 11.4i)T - 97iT^{2} \)
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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

−12.12474037435427452536498945035, −10.82825961910147325824246140894, −9.823629364433393330586829644864, −8.778371901545035127587350364583, −8.077474957117314236441524635974, −7.31141053764044350210915263702, −5.77576963739249303340310975974, −4.30606154926733553653993296690, −3.55340829997501906712883013235, −1.36697038135112943572058897073, 1.64478960200774071026073658143, 3.56040158149008228199444640576, 4.71140210073848463883116659731, 5.69791944066139452494486828151, 7.14085633002392676864889334920, 8.471968058851744102894213813973, 9.093472530926430025371687244804, 9.727371451102350446417809944942, 11.12086093882488472434810410698, 11.67256758499585655700011823611

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