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Properties

Label 2-546-7.2-c1-0-11
Degree $2$
Conductor $546$
Sign $0.827 + 0.561i$
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 + (1.70 − 2.95i)5-s − 0.999·6-s + (−2.62 − 0.358i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1.70 + 2.95i)10-s + (−1.20 − 2.09i)11-s + (0.499 − 0.866i)12-s + 13-s + (1.62 − 2.09i)14-s + 3.41·15-s + (−0.5 + 0.866i)16-s + (−0.207 − 0.358i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.763 − 1.32i)5-s − 0.408·6-s + (−0.990 − 0.135i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.539 + 0.935i)10-s + (−0.363 − 0.630i)11-s + (0.144 − 0.249i)12-s + 0.277·13-s + (0.433 − 0.558i)14-s + 0.881·15-s + (−0.125 + 0.216i)16-s + (−0.0502 − 0.0870i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14555 - 0.351993i\)
\(L(\frac12)\) \(\approx\) \(1.14555 - 0.351993i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.358i)T \)
13 \( 1 - T \)
good5 \( 1 + (-1.70 + 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.20 + 2.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.207 + 0.358i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.91 + 6.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.82T + 29T^{2} \)
31 \( 1 + (4.24 + 7.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.707 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.74 - 6.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.03 - 10.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.792 + 1.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.91 - 3.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 + (-0.707 - 1.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.171 + 0.297i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.65T + 83T^{2} \)
89 \( 1 + (2.70 - 4.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−10.36454875844821864801184639080, −9.464833220074767891975668737576, −9.120659607209273193068872073594, −8.312406822075543958158419757518, −7.13469272911585957439219750214, −5.96557413864121766588461951461, −5.28817221950654261135609917404, −4.24438343095263378539646845021, −2.73333709060986727055704477443, −0.77100527401154253500091032999, 1.76391692566438613686580470264, 2.87427955870346443719974111104, 3.58359567390482365567537781746, 5.52503103610921198378798622234, 6.54819446846114042386064264911, 7.20605523943299968681852092518, 8.253506012854696433634111054484, 9.464983815056076919606691084353, 10.02404579486644668925534200439, 10.63002990687503907227079425003

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