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Properties

Label 2-546-91.54-c1-0-13
Degree $2$
Conductor $546$
Sign $-0.459 + 0.888i$
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 + (−0.981 + 0.263i)5-s + (0.965 − 0.258i)6-s + (2.09 − 1.61i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.508 − 0.880i)10-s + (−4.09 + 1.09i)11-s + (−0.500 + 0.866i)12-s + (0.709 − 3.53i)13-s + (−0.335 + 2.62i)14-s + (0.981 + 0.263i)15-s − 1.00·16-s − 0.895·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.499 − 0.288i)3-s − 0.500i·4-s + (−0.439 + 0.117i)5-s + (0.394 − 0.105i)6-s + (0.791 − 0.611i)7-s + (0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (0.160 − 0.278i)10-s + (−1.23 + 0.330i)11-s + (−0.144 + 0.250i)12-s + (0.196 − 0.980i)13-s + (−0.0896 + 0.701i)14-s + (0.253 + 0.0679i)15-s − 0.250·16-s − 0.217·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.459 + 0.888i$
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: \((2,\ 546,\ (\ :1/2),\ -0.459 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.214046 - 0.351850i\)
\(L(\frac12)\) \(\approx\) \(0.214046 - 0.351850i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 + 1.61i)T \)
13 \( 1 + (-0.709 + 3.53i)T \)
good5 \( 1 + (0.981 - 0.263i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (4.09 - 1.09i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 0.895T + 17T^{2} \)
19 \( 1 + (0.745 - 2.78i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 1.11iT - 23T^{2} \)
29 \( 1 + (4.64 + 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.04 + 3.90i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (6.83 + 6.83i)T + 37iT^{2} \)
41 \( 1 + (-0.375 + 1.39i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.62 + 2.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.821 + 3.06i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.89 + 3.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.46 - 5.46i)T - 59iT^{2} \)
61 \( 1 + (-2.04 + 1.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.11 - 4.17i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.229 - 0.858i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.449 - 0.120i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.14 + 5.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.53 - 7.53i)T + 83iT^{2} \)
89 \( 1 + (9.28 - 9.28i)T - 89iT^{2} \)
97 \( 1 + (-8.06 + 2.16i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.55769210393102689384451410629, −9.812279984496114371961703922620, −8.319934904167125810728496677208, −7.79456557928473304610372084206, −7.19623246645122347933943718704, −5.87420537827919810034389355856, −5.14149336729135791065603406124, −3.92360519357575147605821888444, −2.05366401186661277060402820824, −0.29092082073165477067553533545, 1.75133630526168564403814379197, 3.17556288415809548920798016581, 4.55510269567204503347095115233, 5.29526343295846731740149735699, 6.62846265694792783081138642971, 7.75233137479057571252942095923, 8.554903956315666147280747944339, 9.288002805624707721803319210286, 10.43452494590436453550624651015, 11.10097819407974797410790311330

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