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Properties

Label 2-546-21.5-c1-0-18
Degree $2$
Conductor $546$
Sign $0.435 - 0.900i$
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  + (0.866 + 0.5i)2-s + (1.55 + 0.764i)3-s + (0.499 + 0.866i)4-s + (0.698 − 1.20i)5-s + (0.963 + 1.43i)6-s + (0.352 + 2.62i)7-s + 0.999i·8-s + (1.82 + 2.37i)9-s + (1.20 − 0.698i)10-s + (−1.61 + 0.930i)11-s + (0.114 + 1.72i)12-s − i·13-s + (−1.00 + 2.44i)14-s + (2.01 − 1.34i)15-s + (−0.5 + 0.866i)16-s + (−3.48 − 6.04i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.897 + 0.441i)3-s + (0.249 + 0.433i)4-s + (0.312 − 0.540i)5-s + (0.393 + 0.587i)6-s + (0.133 + 0.991i)7-s + 0.353i·8-s + (0.609 + 0.792i)9-s + (0.382 − 0.220i)10-s + (−0.485 + 0.280i)11-s + (0.0330 + 0.498i)12-s − 0.277i·13-s + (−0.268 + 0.654i)14-s + (0.519 − 0.347i)15-s + (−0.125 + 0.216i)16-s + (−0.845 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.435 - 0.900i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.435 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36429 + 1.48284i\)
\(L(\frac12)\) \(\approx\) \(2.36429 + 1.48284i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.55 - 0.764i)T \)
7 \( 1 + (-0.352 - 2.62i)T \)
13 \( 1 + iT \)
good5 \( 1 + (-0.698 + 1.20i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.61 - 0.930i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.48 + 6.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.37 - 0.794i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.39 + 0.803i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.27iT - 29T^{2} \)
31 \( 1 + (-8.07 + 4.66i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.07 + 3.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.67T + 41T^{2} \)
43 \( 1 + 2.25T + 43T^{2} \)
47 \( 1 + (3.52 - 6.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.86 - 1.07i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.98 + 3.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.3 + 5.98i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.79 - 3.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 + (-7.45 + 4.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.95 - 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.15T + 83T^{2} \)
89 \( 1 + (-3.50 + 6.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−11.04577190450964327940120029324, −9.732762592473517139843384836306, −9.217147258924494633651452260756, −8.255163748789879896356021996524, −7.55377574218044802915464294282, −6.23558494615384773955280197767, −5.09778165581361834168097522530, −4.57119823600744127943858895918, −3.04768679329536563849870976952, −2.19747665031840299270653679335, 1.48234277695261467962186192764, 2.73176828055575619550875014237, 3.71013961943999670313245267866, 4.70854018898169413312307736155, 6.31262682038569109689467546220, 6.85317648909535047631225190675, 7.951047095076352324571900707474, 8.786974772936957211995009990243, 10.13182437957074814341591029672, 10.46837621471842861763411072397

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