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Properties

Label 2-546-91.54-c1-0-4
Degree $2$
Conductor $546$
Sign $-0.787 - 0.616i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  + (−0.707 + 0.707i)2-s + (0.866 + 0.5i)3-s − 1.00i·4-s + (−0.622 + 0.166i)5-s + (−0.965 + 0.258i)6-s + (−1.49 + 2.18i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.322 − 0.558i)10-s + (−0.187 + 0.0502i)11-s + (0.500 − 0.866i)12-s + (2.76 + 2.31i)13-s + (−0.491 − 2.59i)14-s + (−0.622 − 0.166i)15-s − 1.00·16-s − 4.66·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.499 + 0.288i)3-s − 0.500i·4-s + (−0.278 + 0.0746i)5-s + (−0.394 + 0.105i)6-s + (−0.563 + 0.826i)7-s + (0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (0.101 − 0.176i)10-s + (−0.0565 + 0.0151i)11-s + (0.144 − 0.250i)12-s + (0.765 + 0.643i)13-s + (−0.131 − 0.694i)14-s + (−0.160 − 0.0430i)15-s − 0.250·16-s − 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.787 - 0.616i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.787 - 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.298800 + 0.866066i\)
\(L(\frac12)\) \(\approx\) \(0.298800 + 0.866066i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (1.49 - 2.18i)T \)
13 \( 1 + (-2.76 - 2.31i)T \)
good5 \( 1 + (0.622 - 0.166i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.187 - 0.0502i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 4.66T + 17T^{2} \)
19 \( 1 + (0.181 - 0.675i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 5.88iT - 23T^{2} \)
29 \( 1 + (3.99 + 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.500 - 1.86i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-8.11 - 8.11i)T + 37iT^{2} \)
41 \( 1 + (3.14 - 11.7i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.75 - 3.32i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.07 + 4.00i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.01 + 10.4i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.72 - 2.72i)T - 59iT^{2} \)
61 \( 1 + (-3.53 + 2.04i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0733 + 0.273i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.34 - 8.74i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.854 + 0.228i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.95 + 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.79 - 3.79i)T + 83iT^{2} \)
89 \( 1 + (-3.83 + 3.83i)T - 89iT^{2} \)
97 \( 1 + (-6.54 + 1.75i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.25292987042504641578658272028, −9.791490970285041300804906876347, −9.432900935708347068647059194680, −8.498968040369415617817145825918, −7.78650838836032160438822625945, −6.60258767868474444726702827004, −5.87108633293600455042474697457, −4.54835262793321727179099845312, −3.36774893252557288060967191407, −1.95666621421704067323010641136, 0.57622128376045064365601903037, 2.26819564788153739582694229260, 3.51836071456305502307563577624, 4.32910052931405429692313401149, 6.07230278414827004074799823207, 7.09439073808606275974439325823, 7.85419913915292866320251858865, 8.797733137749041250443192698880, 9.451142538372237003109019965708, 10.74453755935409626544289098865

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