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Properties

Label 2-546-91.6-c1-0-8
Degree $2$
Conductor $546$
Sign $0.125 - 0.992i$
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  + (0.258 + 0.965i)2-s + (0.866 + 0.5i)3-s + (−0.866 + 0.499i)4-s + (−0.0374 − 0.0374i)5-s + (−0.258 + 0.965i)6-s + (2.56 + 0.636i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (0.0264 − 0.0458i)10-s + (1.73 − 0.465i)11-s − 12-s + (2.12 − 2.91i)13-s + (0.0501 + 2.64i)14-s + (−0.0137 − 0.0511i)15-s + (0.500 − 0.866i)16-s + (3.26 + 5.65i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.499 + 0.288i)3-s + (−0.433 + 0.249i)4-s + (−0.0167 − 0.0167i)5-s + (−0.105 + 0.394i)6-s + (0.970 + 0.240i)7-s + (−0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (0.00837 − 0.0145i)10-s + (0.523 − 0.140i)11-s − 0.288·12-s + (0.589 − 0.807i)13-s + (0.0133 + 0.706i)14-s + (−0.00353 − 0.0132i)15-s + (0.125 − 0.216i)16-s + (0.791 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.125 - 0.992i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.125 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50320 + 1.32450i\)
\(L(\frac12)\) \(\approx\) \(1.50320 + 1.32450i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-2.56 - 0.636i)T \)
13 \( 1 + (-2.12 + 2.91i)T \)
good5 \( 1 + (0.0374 + 0.0374i)T + 5iT^{2} \)
11 \( 1 + (-1.73 + 0.465i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.26 - 5.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.60 - 5.97i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (6.84 + 3.94i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.18 - 5.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.81 - 1.81i)T + 31iT^{2} \)
37 \( 1 + (4.19 - 1.12i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.982 - 0.263i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9.47 + 5.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.86 + 4.86i)T - 47iT^{2} \)
53 \( 1 + 4.52T + 53T^{2} \)
59 \( 1 + (9.90 + 2.65i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.13 + 1.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.84 + 6.87i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (10.4 + 2.80i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.17 + 3.17i)T - 73iT^{2} \)
79 \( 1 - 5.14T + 79T^{2} \)
83 \( 1 + (-6.12 - 6.12i)T + 83iT^{2} \)
89 \( 1 + (1.38 + 5.15i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.92 + 14.6i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.68399322915000526404682203865, −10.24489795274592588558208882078, −8.855959607947559715181265584562, −8.247867085941746026189852355962, −7.75135768711984748270994931662, −6.24625655533716045554577886048, −5.59177972732074649686233896182, −4.30492731346881059522123015878, −3.52797746814845481970021382623, −1.75010809228634705673086010977, 1.27049497441862939327151537478, 2.43515183522537690999660356107, 3.80387899094444912020883902891, 4.64969064578005040904525039293, 5.87648057729623576134901710292, 7.19515227437809851021730785602, 7.915652132762652167406275539971, 9.128892398178067078975011135494, 9.507382558360427094733450219668, 10.82655702523031080127469466723

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