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Properties

Label 2-546-21.17-c1-0-31
Degree $2$
Conductor $546$
Sign $-0.988 + 0.152i$
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  + (0.866 − 0.5i)2-s + (0.110 − 1.72i)3-s + (0.499 − 0.866i)4-s + (−1.50 − 2.61i)5-s + (−0.768 − 1.55i)6-s + (0.237 + 2.63i)7-s − 0.999i·8-s + (−2.97 − 0.381i)9-s + (−2.61 − 1.50i)10-s + (−4.75 − 2.74i)11-s + (−1.44 − 0.959i)12-s + i·13-s + (1.52 + 2.16i)14-s + (−4.68 + 2.32i)15-s + (−0.5 − 0.866i)16-s + (−0.598 + 1.03i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.0636 − 0.997i)3-s + (0.249 − 0.433i)4-s + (−0.674 − 1.16i)5-s + (−0.313 − 0.633i)6-s + (0.0896 + 0.995i)7-s − 0.353i·8-s + (−0.991 − 0.127i)9-s + (−0.826 − 0.477i)10-s + (−1.43 − 0.827i)11-s + (−0.416 − 0.277i)12-s + 0.277i·13-s + (0.407 + 0.578i)14-s + (−1.20 + 0.599i)15-s + (−0.125 − 0.216i)16-s + (−0.145 + 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.988 + 0.152i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.988 + 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108451 - 1.41718i\)
\(L(\frac12)\) \(\approx\) \(0.108451 - 1.41718i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.110 + 1.72i)T \)
7 \( 1 + (-0.237 - 2.63i)T \)
13 \( 1 - iT \)
good5 \( 1 + (1.50 + 2.61i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.75 + 2.74i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.598 - 1.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.49 + 3.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.22 + 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.12iT - 29T^{2} \)
31 \( 1 + (-4.87 - 2.81i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.76 + 8.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.26T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + (3.89 + 6.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.53 - 3.77i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.06 + 3.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.26 + 4.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.97 + 8.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.25iT - 71T^{2} \)
73 \( 1 + (0.501 + 0.289i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.649 - 1.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (-2.75 - 4.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.59iT - 97T^{2} \)
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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

−10.75329099325744454345767211578, −9.244493028372294536437796809171, −8.480505207197179217629456210988, −7.86148357079202581238247518274, −6.64089149395778313710557027091, −5.38242563761915035203259731446, −5.05343984895037846624012779564, −3.32516349558560822540930526137, −2.27376470462580232126199815454, −0.65537486074878087558329962084, 2.85723545944781252801920491118, 3.51712560485836088423062477015, 4.62144637347134934800653968505, 5.40565101755428390617749718862, 6.83077518065436980760055884260, 7.54136520969090749213892279727, 8.225955072339682320537087164400, 9.938242682465954594636210938355, 10.26770516268268044364219200916, 11.19480808558394644213983641344

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