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Properties

Label 2-273-91.73-c1-0-10
Degree $2$
Conductor $273$
Sign $0.984 - 0.172i$
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  + (0.0455 − 0.170i)2-s + (0.866 + 0.5i)3-s + (1.70 + 0.984i)4-s + (−0.317 − 0.0851i)5-s + (0.124 − 0.124i)6-s + (1.64 − 2.06i)7-s + (0.494 − 0.494i)8-s + (0.499 + 0.866i)9-s + (−0.0289 + 0.0501i)10-s + (−0.952 − 3.55i)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  + (0.0322 − 0.120i)2-s + (0.499 + 0.288i)3-s + (0.852 + 0.492i)4-s + (−0.142 − 0.0380i)5-s + (0.0508 − 0.0508i)6-s + (0.623 − 0.782i)7-s + (0.174 − 0.174i)8-s + (0.166 + 0.288i)9-s + (−0.00916 + 0.0158i)10-s + (−0.287 − 1.07i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.984 - 0.172i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.984 - 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77775 + 0.154875i\)
\(L(\frac12)\) \(\approx\) \(1.77775 + 0.154875i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 2.06i)T \)
13 \( 1 + (1.90 - 3.06i)T \)
good2 \( 1 + (-0.0455 + 0.170i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (0.317 + 0.0851i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.952 + 3.55i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.19 - 3.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.13 - 0.839i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.57 + 1.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.47T + 29T^{2} \)
31 \( 1 + (0.540 + 2.01i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.29 - 0.614i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.39 + 1.39i)T - 41iT^{2} \)
43 \( 1 - 0.148iT - 43T^{2} \)
47 \( 1 + (1.29 - 4.83i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.51 + 9.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.71 - 2.60i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.35 - 1.93i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.99 - 1.60i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.84 + 5.84i)T + 71iT^{2} \)
73 \( 1 + (4.90 - 1.31i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.57 + 2.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.02 + 4.02i)T - 83iT^{2} \)
89 \( 1 + (2.92 - 10.9i)T + (-77.0 - 44.5i)T^{2} \)
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

−11.64586358616204322830407960558, −11.10989996227208693609909182703, −10.23345424623431831107536311904, −8.946504017902508063440226616128, −7.927918457758723123290345310083, −7.29814786695967659705005837484, −5.99325065051528424831790344195, −4.37192114191180475971434182168, −3.41025621685539890480544995563, −1.93532973467821001798834890248, 1.86148990405269980339417236110, 2.90681232036544420015694972321, 4.90074004172294478166160423002, 5.78295914772098536783158557347, 7.39894402530285512945950155343, 7.51571192176437598537607519523, 9.087120938640203858566789177909, 9.892893345139855341349740182442, 11.10061914639580941667314626675, 11.81912215935829712066640558042

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