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Properties

Label 2-546-91.9-c1-0-14
Degree $2$
Conductor $546$
Sign $-0.347 + 0.937i$
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 − 3-s + (−0.499 + 0.866i)4-s + (2.05 − 3.56i)5-s + (0.5 + 0.866i)6-s + (1.65 − 2.06i)7-s + 0.999·8-s + 9-s − 4.11·10-s + 4.04·11-s + (0.499 − 0.866i)12-s + (1.81 + 3.11i)13-s + (−2.61 − 0.405i)14-s + (−2.05 + 3.56i)15-s + (−0.5 − 0.866i)16-s + (−0.357 + 0.619i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s + (0.920 − 1.59i)5-s + (0.204 + 0.353i)6-s + (0.626 − 0.779i)7-s + 0.353·8-s + 0.333·9-s − 1.30·10-s + 1.21·11-s + (0.144 − 0.249i)12-s + (0.503 + 0.864i)13-s + (−0.698 − 0.108i)14-s + (−0.531 + 0.920i)15-s + (−0.125 − 0.216i)16-s + (−0.0867 + 0.150i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.347 + 0.937i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.347 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.740976 - 1.06505i\)
\(L(\frac12)\) \(\approx\) \(0.740976 - 1.06505i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (-1.65 + 2.06i)T \)
13 \( 1 + (-1.81 - 3.11i)T \)
good5 \( 1 + (-2.05 + 3.56i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 4.04T + 11T^{2} \)
17 \( 1 + (0.357 - 0.619i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.84T + 19T^{2} \)
23 \( 1 + (-2.04 - 3.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.50 + 7.80i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.82 - 3.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.59 + 6.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.88 - 4.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.28 - 2.22i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.28 + 2.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.35 + 2.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.45 - 11.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 9.43T + 61T^{2} \)
67 \( 1 + 1.16T + 67T^{2} \)
71 \( 1 + (-5.10 - 8.83i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.25 + 2.17i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.70 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + (-5.45 - 9.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.10 + 5.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.50684730601328156302921189237, −9.622940842374215736500912358156, −8.938861838843357161962985583460, −8.219195848856013008857855746943, −6.81426150688279351370929014094, −5.83301128289409746348489370247, −4.58728335245424299646857759940, −4.13518349786327734910885763731, −1.77220443847729937085148007749, −1.06546948865564173084637249708, 1.71973548311791833033901727978, 3.13145363097081387844168769878, 4.80656489548400298478818178127, 5.96871646563844632532101914456, 6.39184096589584129451880129858, 7.19120199505972492722074334319, 8.463817834899467088789296097694, 9.278319977460875998769120378175, 10.40325928765564763488420793417, 10.76594088939850563839437630370

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