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Properties

Label 2-546-91.23-c1-0-17
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  − i·2-s + (−0.5 + 0.866i)3-s − 4-s + (−2.21 − 1.27i)5-s + (0.866 + 0.5i)6-s + (1.74 − 1.99i)7-s + i·8-s + (−0.499 − 0.866i)9-s + (−1.27 + 2.21i)10-s + (−0.449 − 0.259i)11-s + (0.5 − 0.866i)12-s + (−1.57 + 3.24i)13-s + (−1.99 − 1.74i)14-s + (2.21 − 1.27i)15-s + 16-s − 0.868·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.288 + 0.499i)3-s − 0.5·4-s + (−0.988 − 0.570i)5-s + (0.353 + 0.204i)6-s + (0.658 − 0.752i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.403 + 0.699i)10-s + (−0.135 − 0.0782i)11-s + (0.144 − 0.249i)12-s + (−0.436 + 0.899i)13-s + (−0.532 − 0.465i)14-s + (0.570 − 0.329i)15-s + 0.250·16-s − 0.210·17-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: $\chi_{546} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0392636 + 0.127884i\)
\(L(\frac12)\) \(\approx\) \(0.0392636 + 0.127884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-1.74 + 1.99i)T \)
13 \( 1 + (1.57 - 3.24i)T \)
good5 \( 1 + (2.21 + 1.27i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.449 + 0.259i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.868T + 17T^{2} \)
19 \( 1 + (4.00 - 2.31i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.52T + 23T^{2} \)
29 \( 1 + (1.98 + 3.43i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.49 - 2.01i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.92iT - 37T^{2} \)
41 \( 1 + (5.87 - 3.39i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.24 + 3.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.81 + 1.62i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.12 + 7.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.40iT - 59T^{2} \)
61 \( 1 + (-3.01 - 5.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.54 - 5.51i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.47 - 2.58i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-10.3 + 5.97i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.47 + 9.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.9iT - 83T^{2} \)
89 \( 1 + 8.02iT - 89T^{2} \)
97 \( 1 + (13.0 + 7.51i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.36332812001503680304975653379, −9.643367252744646660642117059709, −8.422866745418921303756538038174, −7.958084268739617692036961048650, −6.63547697100756161414687194389, −5.17533932651092275039034493181, −4.24546135165079579776751941777, −3.83558195308605492103444473913, −1.88978418161977609051561169910, −0.07732112344857346577820794031, 2.27350855591096478929266042353, 3.78829742209017349415505463256, 4.99916113011866587795290825192, 5.89999520815515527468547902534, 6.90783046295835784260028466569, 7.83890872597324721058327490610, 8.192832623520197346001684088273, 9.371831447934265720970580614618, 10.66948639554344658845032182265, 11.29425756299387748031473221208

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