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Properties

Label 2-546-39.32-c1-0-13
Degree $2$
Conductor $546$
Sign $0.549 - 0.835i$
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 + (0.505 + 1.65i)3-s + (0.866 − 0.499i)4-s + (2.95 + 2.95i)5-s + (0.916 + 1.46i)6-s + (0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (−2.48 + 1.67i)9-s + (3.62 + 2.09i)10-s + (0.182 + 0.679i)11-s + (1.26 + 1.18i)12-s + (1.11 − 3.42i)13-s − i·14-s + (−3.40 + 6.39i)15-s + (0.500 − 0.866i)16-s + (−2.06 − 3.56i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.291 + 0.956i)3-s + (0.433 − 0.249i)4-s + (1.32 + 1.32i)5-s + (0.374 + 0.599i)6-s + (0.0978 − 0.365i)7-s + (0.249 − 0.249i)8-s + (−0.829 + 0.557i)9-s + (1.14 + 0.661i)10-s + (0.0549 + 0.204i)11-s + (0.365 + 0.341i)12-s + (0.310 − 0.950i)13-s − 0.267i·14-s + (−0.879 + 1.65i)15-s + (0.125 − 0.216i)16-s + (−0.499 − 0.865i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.549 - 0.835i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.549 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44287 + 1.31740i\)
\(L(\frac12)\) \(\approx\) \(2.44287 + 1.31740i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.505 - 1.65i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-1.11 + 3.42i)T \)
good5 \( 1 + (-2.95 - 2.95i)T + 5iT^{2} \)
11 \( 1 + (-0.182 - 0.679i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.06 + 3.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.67 + 2.05i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.12 + 1.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.72 - 0.994i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.11 + 2.11i)T - 31iT^{2} \)
37 \( 1 + (-4.55 + 1.22i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.49 + 0.400i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (10.5 - 6.08i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.65 + 3.65i)T - 47iT^{2} \)
53 \( 1 - 0.161iT - 53T^{2} \)
59 \( 1 + (-5.46 - 1.46i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.568 - 0.985i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.89 - 10.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.345 + 1.28i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.56 - 4.56i)T + 73iT^{2} \)
79 \( 1 - 7.36T + 79T^{2} \)
83 \( 1 + (10.3 + 10.3i)T + 83iT^{2} \)
89 \( 1 + (-2.00 - 7.47i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (16.1 + 4.34i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.79656829699369432499760641232, −10.24978311021137263443355608815, −9.567462087217779808379666837121, −8.406647910409047887349865417637, −6.99644943410814485651806990189, −6.24268042884026391472507706661, −5.29215964226382716884458830793, −4.23551810565082346282275522568, −2.97999559996594820471301807384, −2.32465358813610039980928084591, 1.56271518815734746569767560962, 2.29121169926702262223364719643, 4.08950860875603546448240603595, 5.20857436065604586066739540031, 6.20639900914395234237306546857, 6.54960397554572776771413924053, 8.254005867359552444206253492953, 8.636729531818311757008575123849, 9.528563339562362412395484506662, 10.81490992598029119377733519787

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