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Properties

Label 2-546-91.33-c1-0-6
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: $\chi_{546} (397, \cdot )$
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

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

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