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Properties

Label 2-546-273.17-c1-0-10
Degree $2$
Conductor $546$
Sign $0.754 - 0.656i$
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  − 2-s + (−1.5 + 0.866i)3-s + 4-s + (3 − 1.73i)5-s + (1.5 − 0.866i)6-s + (−0.5 + 2.59i)7-s − 8-s + (1.5 − 2.59i)9-s + (−3 + 1.73i)10-s + (3 + 5.19i)11-s + (−1.5 + 0.866i)12-s + (−2.5 − 2.59i)13-s + (0.5 − 2.59i)14-s + (−3 + 5.19i)15-s + 16-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.866 + 0.499i)3-s + 0.5·4-s + (1.34 − 0.774i)5-s + (0.612 − 0.353i)6-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (0.5 − 0.866i)9-s + (−0.948 + 0.547i)10-s + (0.904 + 1.56i)11-s + (−0.433 + 0.249i)12-s + (−0.693 − 0.720i)13-s + (0.133 − 0.694i)14-s + (−0.774 + 1.34i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.754 - 0.656i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.754 - 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945841 + 0.353649i\)
\(L(\frac12)\) \(\approx\) \(0.945841 + 0.353649i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (0.5 - 2.59i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good5 \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + (-6 - 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.66iT - 37T^{2} \)
41 \( 1 + (-9 - 5.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6 + 3.46i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 1.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + 3.46iT - 89T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.56230978220468315845599432778, −9.843691173542517022588394941901, −9.362358875780042720200944838944, −8.712119187393681288506658530560, −7.06738395159695094126951902046, −6.33561102252767572507756785780, −5.29053477758604399199797271050, −4.72821050387656558243373913926, −2.63357097597375079113730642612, −1.29881398216898491119744554424, 0.977735717229104235874247603134, 2.25939069747099897410742993062, 3.85227911136320157557543189351, 5.69357377689222093803945954762, 6.16723364235994099169436458262, 7.00967729811912764498299557089, 7.77214652439216223516170469366, 9.255990877082562062631749253532, 9.875339453351827224945861881078, 10.73378141141859610697875123829

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