Properties

Label 2-546-91.80-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.942 + 0.333i$
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 + i·3-s + (−0.866 + 0.499i)4-s + (−0.231 + 0.864i)5-s + (−0.965 + 0.258i)6-s + (−0.401 + 2.61i)7-s + (−0.707 − 0.707i)8-s − 9-s − 0.894·10-s + (−3.46 − 3.46i)11-s + (−0.499 − 0.866i)12-s + (−2.86 + 2.19i)13-s + (−2.62 + 0.289i)14-s + (−0.864 − 0.231i)15-s + (0.500 − 0.866i)16-s + (1.34 + 2.32i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + 0.577i·3-s + (−0.433 + 0.249i)4-s + (−0.103 + 0.386i)5-s + (−0.394 + 0.105i)6-s + (−0.151 + 0.988i)7-s + (−0.249 − 0.249i)8-s − 0.333·9-s − 0.282·10-s + (−1.04 − 1.04i)11-s + (−0.144 − 0.249i)12-s + (−0.794 + 0.607i)13-s + (−0.702 + 0.0773i)14-s + (−0.223 − 0.0598i)15-s + (0.125 − 0.216i)16-s + (0.325 + 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.942 + 0.333i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148746 - 0.866057i\)
\(L(\frac12)\) \(\approx\) \(0.148746 - 0.866057i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (0.401 - 2.61i)T \)
13 \( 1 + (2.86 - 2.19i)T \)
good5 \( 1 + (0.231 - 0.864i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.46 + 3.46i)T + 11iT^{2} \)
17 \( 1 + (-1.34 - 2.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.36 + 2.36i)T + 19iT^{2} \)
23 \( 1 + (-5.93 - 3.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.14 + 1.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.73 - 2.07i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (7.37 - 1.97i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.707 + 2.64i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.45 + 2.57i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-12.1 - 3.25i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.53 - 7.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.26 - 0.876i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 + (-9.47 + 9.47i)T - 67iT^{2} \)
71 \( 1 + (2.94 + 10.9i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.52 - 13.1i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.28 - 5.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.49 - 2.49i)T + 83iT^{2} \)
89 \( 1 + (-2.63 - 9.83i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.09 + 0.830i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.12276696897975500575167597692, −10.48890192304700056362490248585, −9.184108690559697518650928551865, −8.808139116281779398386420828693, −7.67410159702173846184267305167, −6.72022131635397151194613667688, −5.56377619292995878737833616840, −5.08514308926656714495826726227, −3.60272927736187056507838287686, −2.61788916371400805462850319226, 0.45833554796862092414698810529, 2.06681231651631647301010392795, 3.27989822466292979746521427112, 4.65560065973477651305877456456, 5.32174900211714789426380003611, 6.91599496494385173190836403247, 7.52960335794523999071677316292, 8.527012895141868090936404910400, 9.689961166701551095415529314212, 10.41358900595923480073190815977

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