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Properties

Label 2-546-91.41-c1-0-13
Degree $2$
Conductor $546$
Sign $0.796 + 0.604i$
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  + (−0.965 − 0.258i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.499i)4-s + (2.53 + 2.53i)5-s + (−0.965 + 0.258i)6-s + (−1.06 − 2.42i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−1.78 − 3.09i)10-s + (1.24 − 4.66i)11-s + 12-s + (−3.47 + 0.965i)13-s + (0.400 + 2.61i)14-s + (3.45 + 0.926i)15-s + (0.500 + 0.866i)16-s + (3.64 − 6.31i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 + 0.249i)4-s + (1.13 + 1.13i)5-s + (−0.394 + 0.105i)6-s + (−0.402 − 0.915i)7-s + (−0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.565 − 0.980i)10-s + (0.376 − 1.40i)11-s + 0.288·12-s + (−0.963 + 0.267i)13-s + (0.107 + 0.698i)14-s + (0.892 + 0.239i)15-s + (0.125 + 0.216i)16-s + (0.883 − 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.796 + 0.604i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.796 + 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39895 - 0.471148i\)
\(L(\frac12)\) \(\approx\) \(1.39895 - 0.471148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (1.06 + 2.42i)T \)
13 \( 1 + (3.47 - 0.965i)T \)
good5 \( 1 + (-2.53 - 2.53i)T + 5iT^{2} \)
11 \( 1 + (-1.24 + 4.66i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.64 + 6.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.46 + 1.46i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.69 - 1.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.05 - 5.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.07 - 6.07i)T + 31iT^{2} \)
37 \( 1 + (2.09 - 7.83i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.957 + 3.57i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.401 + 0.231i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.23 + 3.23i)T - 47iT^{2} \)
53 \( 1 + 5.63T + 53T^{2} \)
59 \( 1 + (2.29 + 8.55i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.62 + 2.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.63 - 1.24i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.02 - 3.82i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.575 - 0.575i)T - 73iT^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (-7.77 - 7.77i)T + 83iT^{2} \)
89 \( 1 + (-4.21 - 1.12i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (14.8 - 3.99i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.42232376922185948642872210566, −9.810918275031986465612998045444, −9.256107765357737040308180430759, −7.974023632329410489565893175441, −7.00570810090606152825543449635, −6.62353420390639882299969541573, −5.29766314900322430542893917918, −3.25044631323831808542445766084, −2.85434855639975158567065411662, −1.15618239043755609213640655766, 1.59848362023730083636376218903, 2.55562763587137487496585505880, 4.38413579196906890347938062345, 5.49489865339813638598147145459, 6.17389003824286948318574282059, 7.62330700068436617431628816180, 8.363557076258067286862329561492, 9.463257491516078504258527184154, 9.651945641026892049072939736256, 10.27189124464836300686367414026

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