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Properties

Label 2-546-21.5-c1-0-3
Degree $2$
Conductor $546$
Sign $0.966 - 0.257i$
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 + (−0.631 − 1.61i)3-s + (0.499 + 0.866i)4-s + (−1.49 + 2.59i)5-s + (−0.259 + 1.71i)6-s + (−0.0366 − 2.64i)7-s − 0.999i·8-s + (−2.20 + 2.03i)9-s + (2.59 − 1.49i)10-s + (−1.10 + 0.638i)11-s + (1.08 − 1.35i)12-s + i·13-s + (−1.29 + 2.30i)14-s + (5.12 + 0.776i)15-s + (−0.5 + 0.866i)16-s + (−0.544 − 0.942i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.364 − 0.931i)3-s + (0.249 + 0.433i)4-s + (−0.669 + 1.16i)5-s + (−0.105 + 0.699i)6-s + (−0.0138 − 0.999i)7-s − 0.353i·8-s + (−0.733 + 0.679i)9-s + (0.820 − 0.473i)10-s + (−0.333 + 0.192i)11-s + (0.312 − 0.390i)12-s + 0.277i·13-s + (−0.345 + 0.617i)14-s + (1.32 + 0.200i)15-s + (−0.125 + 0.216i)16-s + (−0.132 − 0.228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.966 - 0.257i$
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: \((2,\ 546,\ (\ :1/2),\ 0.966 - 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687210 + 0.0899887i\)
\(L(\frac12)\) \(\approx\) \(0.687210 + 0.0899887i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.631 + 1.61i)T \)
7 \( 1 + (0.0366 + 2.64i)T \)
13 \( 1 - iT \)
good5 \( 1 + (1.49 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.10 - 0.638i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.544 + 0.942i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.52 - 3.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.47 - 3.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.01iT - 29T^{2} \)
31 \( 1 + (7.20 - 4.15i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.363 + 0.629i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.09T + 41T^{2} \)
43 \( 1 - 8.73T + 43T^{2} \)
47 \( 1 + (0.962 - 1.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.80 + 5.08i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.18 - 2.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.29 + 3.05i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.68 - 9.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.18iT - 71T^{2} \)
73 \( 1 + (9.69 - 5.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.66 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.63T + 83T^{2} \)
89 \( 1 + (6.56 - 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.3iT - 97T^{2} \)
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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

−11.01306730420586928863890729607, −10.23596305007701504626197289261, −9.098132626693559033265442673293, −7.70505363998751029263975698210, −7.38637114875481916581129445892, −6.80233177204087198425653406621, −5.41797363391912569579713983787, −3.75791469090784372514807821135, −2.78937634922315708358115237795, −1.18702450099679857133039665402, 0.61095416418161958801403797021, 2.85255516301211589190314006369, 4.35856826596852685528040614338, 5.26660760781389385251070630149, 5.87105254119505436866851550723, 7.37913027183114169971512065589, 8.379297958377949469796256453316, 9.101403079160744697915335868155, 9.497166884867512458674803361520, 10.81417716970159373221965614843

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