Properties

Label 2-273-91.9-c1-0-9
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

Related objects

Downloads

Learn more

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: $\chi_{273} (100, \cdot )$
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 \( 1 + (1.79 + 3.11i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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