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Properties

Label 2-546-91.80-c1-0-15
Degree $2$
Conductor $546$
Sign $-0.905 + 0.423i$
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 + (0.807 − 3.01i)5-s + (−0.965 + 0.258i)6-s + (1.90 − 1.83i)7-s + (0.707 + 0.707i)8-s − 9-s − 3.11·10-s + (1.08 + 1.08i)11-s + (0.499 + 0.866i)12-s + (0.598 − 3.55i)13-s + (−2.26 − 1.36i)14-s + (−3.01 − 0.807i)15-s + (0.500 − 0.866i)16-s + (1.73 + 3.00i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s − 0.577i·3-s + (−0.433 + 0.249i)4-s + (0.361 − 1.34i)5-s + (−0.394 + 0.105i)6-s + (0.721 − 0.692i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s − 0.986·10-s + (0.326 + 0.326i)11-s + (0.144 + 0.249i)12-s + (0.165 − 0.986i)13-s + (−0.605 − 0.365i)14-s + (−0.777 − 0.208i)15-s + (0.125 − 0.216i)16-s + (0.420 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.905 + 0.423i$
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: \((2,\ 546,\ (\ :1/2),\ -0.905 + 0.423i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.302183 - 1.35842i\)
\(L(\frac12)\) \(\approx\) \(0.302183 - 1.35842i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.90 + 1.83i)T \)
13 \( 1 + (-0.598 + 3.55i)T \)
good5 \( 1 + (-0.807 + 3.01i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.08 - 1.08i)T + 11iT^{2} \)
17 \( 1 + (-1.73 - 3.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.52 + 3.52i)T + 19iT^{2} \)
23 \( 1 + (-6.43 - 3.71i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.30 - 2.25i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (10.1 - 2.72i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.57 - 1.22i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.819 + 3.05i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.51 - 3.76i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.42 + 2.52i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.34 - 5.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.22 - 2.20i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 7.63iT - 61T^{2} \)
67 \( 1 + (-2.12 + 2.12i)T - 67iT^{2} \)
71 \( 1 + (-0.689 - 2.57i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.76 + 6.59i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.33 - 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.39 - 5.39i)T + 83iT^{2} \)
89 \( 1 + (-2.05 - 7.67i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.27 + 1.14i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.63362282433468442550773054884, −9.423610512881690277901691523494, −8.710233834107593555381516449212, −7.976918475326929219612593229982, −6.98582826763707053313057331463, −5.46674742718247559157105213895, −4.79543188309701506837064371387, −3.53133101809570900724004864839, −1.80214238239732909748446447692, −0.937002071836520614704644822823, 2.14751495713674660550226553015, 3.50106918973922783644482985716, 4.75923963644839812560007215492, 5.81782186181953245174807664899, 6.59049050066787043723934858611, 7.49559369982446290726639412554, 8.662262224627953672888442390360, 9.277459045690434812279299644947, 10.28846252472815930641458583613, 11.05579399629526993427854221126

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