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Properties

Label 2-273-91.5-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: $\chi_{273} (187, \cdot )$
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

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

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