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Properties

Label 2-546-273.173-c1-0-34
Degree $2$
Conductor $546$
Sign $-0.197 + 0.980i$
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 + (0.991 + 1.41i)3-s + (−0.499 + 0.866i)4-s + (−3.72 − 2.14i)5-s + (−0.733 + 1.56i)6-s + (−1.08 − 2.41i)7-s − 0.999·8-s + (−1.03 + 2.81i)9-s − 4.29i·10-s − 4.52·11-s + (−1.72 + 0.148i)12-s + (2.01 − 2.98i)13-s + (1.54 − 2.14i)14-s + (−0.639 − 7.41i)15-s + (−0.5 − 0.866i)16-s + (−0.0962 + 0.166i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.572 + 0.819i)3-s + (−0.249 + 0.433i)4-s + (−1.66 − 0.960i)5-s + (−0.299 + 0.640i)6-s + (−0.410 − 0.911i)7-s − 0.353·8-s + (−0.344 + 0.938i)9-s − 1.35i·10-s − 1.36·11-s + (−0.498 + 0.0429i)12-s + (0.559 − 0.828i)13-s + (0.413 − 0.573i)14-s + (−0.165 − 1.91i)15-s + (−0.125 − 0.216i)16-s + (−0.0233 + 0.0404i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.197 + 0.980i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.187434 - 0.229001i\)
\(L(\frac12)\) \(\approx\) \(0.187434 - 0.229001i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.991 - 1.41i)T \)
7 \( 1 + (1.08 + 2.41i)T \)
13 \( 1 + (-2.01 + 2.98i)T \)
good5 \( 1 + (3.72 + 2.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 4.52T + 11T^{2} \)
17 \( 1 + (0.0962 - 0.166i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 1.69T + 19T^{2} \)
23 \( 1 + (-1.22 + 0.704i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.66 + 5.00i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.16 + 2.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.93 - 2.85i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.24 + 0.717i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.73 - 2.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.70 - 5.02i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.59 - 3.22i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.17 - 1.83i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 8.11iT - 61T^{2} \)
67 \( 1 + 5.05iT - 67T^{2} \)
71 \( 1 + (5.59 + 9.68i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.90 - 10.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.93 - 3.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.15iT - 83T^{2} \)
89 \( 1 + (1.87 - 1.08i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.86 + 11.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.66972340319889636819833390292, −9.494472583191072326056998038058, −8.473775862918435690451302183264, −7.88287991417385487757857537307, −7.38391964427461397287395612041, −5.62230731170722265507459574252, −4.67550668789487109023193170939, −3.95123777579058416155618258309, −3.16604347864180635088170201366, −0.13493807144763938057931723416, 2.22888694088001808974006805319, 3.16241226601926959039813056244, 3.90396615948474617066975919828, 5.48984262332437463990076249849, 6.73563008008179406109716607335, 7.42065422268188055755925660148, 8.401721475198996155987220729784, 9.083838108673492908099129630970, 10.51083153857260975522750553153, 11.22688743323035160300006960885

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