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Properties

Label 2-546-13.3-c1-0-10
Degree $2$
Conductor $546$
Sign $0.859 + 0.511i$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 3·5-s + (0.499 − 0.866i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 − 2.59i)10-s + (−2 − 3.46i)11-s − 0.999·12-s + (3.5 − 0.866i)13-s − 0.999·14-s + (1.5 + 2.59i)15-s + (−0.5 − 0.866i)16-s + (2.5 − 4.33i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.34·5-s + (0.204 − 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.474 − 0.821i)10-s + (−0.603 − 1.04i)11-s − 0.288·12-s + (0.970 − 0.240i)13-s − 0.267·14-s + (0.387 + 0.670i)15-s + (−0.125 − 0.216i)16-s + (0.606 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60565 - 0.441288i\)
\(L(\frac12)\) \(\approx\) \(1.60565 - 0.441288i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 7T + 53T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6 - 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9 - 15.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.53343115532845636195354206352, −10.03784542072222927536970703625, −9.014479191310767591713347856317, −8.479032050064001711359886192138, −7.27980534391095969980245568447, −5.86304369308204668490255619496, −5.19928598269149437855311638599, −3.66893432987448413897029189361, −2.74247361109148928325391222304, −1.31391052449536356878880577333, 1.52907911793429289417876431628, 2.55518251274908605510094966011, 4.47035994590382656832482485186, 5.66005246864061023392537503436, 6.32151837875218101606599151865, 7.19512410512835180601112530032, 8.382321443687000781853379714171, 8.865907095545422819853026582045, 10.03446004932540000580444650919, 10.41030222580722124453489354693

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