Properties

Label 2-273-91.81-c1-0-6
Degree $2$
Conductor $273$
Sign $0.958 - 0.286i$
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  + (0.379 − 0.656i)2-s − 3-s + (0.712 + 1.23i)4-s + (−0.357 − 0.619i)5-s + (−0.379 + 0.656i)6-s + (−1.32 + 2.29i)7-s + 2.59·8-s + 9-s − 0.542·10-s + 4.48·11-s + (−0.712 − 1.23i)12-s + (3.26 + 1.52i)13-s + (1.00 + 1.73i)14-s + (0.357 + 0.619i)15-s + (−0.439 + 0.761i)16-s + (1.88 + 3.26i)17-s + ⋯
L(s)  = 1  + (0.268 − 0.464i)2-s − 0.577·3-s + (0.356 + 0.616i)4-s + (−0.160 − 0.277i)5-s + (−0.154 + 0.268i)6-s + (−0.499 + 0.866i)7-s + 0.918·8-s + 0.333·9-s − 0.171·10-s + 1.35·11-s + (−0.205 − 0.356i)12-s + (0.905 + 0.424i)13-s + (0.268 + 0.464i)14-s + (0.0924 + 0.160i)15-s + (−0.109 + 0.190i)16-s + (0.457 + 0.792i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.958 - 0.286i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.958 - 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35069 + 0.197804i\)
\(L(\frac12)\) \(\approx\) \(1.35069 + 0.197804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (1.32 - 2.29i)T \)
13 \( 1 + (-3.26 - 1.52i)T \)
good2 \( 1 + (-0.379 + 0.656i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.357 + 0.619i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
17 \( 1 + (-1.88 - 3.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 5.92T + 19T^{2} \)
23 \( 1 + (-0.465 + 0.806i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.12 + 1.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.191 + 0.331i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.328 + 0.569i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.29 + 3.97i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.50 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.18 + 7.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.21 - 2.10i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.80 + 4.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.99T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + (-2.14 + 3.71i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.34 - 2.32i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.75 - 4.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + (-6.05 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.79 - 3.11i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.01850710438287424075367493698, −11.28508040578802551814814007896, −10.33046288155551496761492749371, −8.991318074549704921013197246084, −8.260894600823920227222352679415, −6.69038175135103773835096678603, −6.12045118312292065095000626777, −4.44949636037069296678038950240, −3.53209386760276090086468455801, −1.83803061368077340385892476346, 1.23101484574105595859377014101, 3.57588641953180242238898302365, 4.76312041047419666298631066080, 6.12063135685197037681874338454, 6.66039991720937733642389817304, 7.54282580417791983955597100104, 9.112916627427768153628052462584, 10.21394385253598975435353041236, 10.90778651762103543017186661977, 11.64958665142258941458388371394

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