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Properties

Label 2-546-91.59-c1-0-16
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: $\chi_{546} (241, \cdot )$
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

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

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