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Properties

Label 2-42e2-21.11-c2-0-13
Degree $2$
Conductor $1764$
Sign $0.875 - 0.483i$
Analytic cond. $48.0655$
Root an. cond. $6.93293$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.08 − 1.77i)5-s + (8.03 + 4.63i)11-s + 4.24·13-s + (2.11 + 1.22i)17-s + (10.3 + 17.9i)19-s + (0.682 − 0.394i)23-s + (−6.17 + 10.6i)25-s − 7.69i·29-s + (11.7 − 20.4i)31-s + (11.6 + 20.2i)37-s + 51.5i·41-s − 49.3·43-s + (−13.6 + 7.88i)47-s + (65.1 + 37.6i)53-s + 32.9·55-s + ⋯
L(s)  = 1  + (0.616 − 0.355i)5-s + (0.730 + 0.421i)11-s + 0.326·13-s + (0.124 + 0.0718i)17-s + (0.546 + 0.945i)19-s + (0.0296 − 0.0171i)23-s + (−0.246 + 0.427i)25-s − 0.265i·29-s + (0.380 − 0.658i)31-s + (0.315 + 0.546i)37-s + 1.25i·41-s − 1.14·43-s + (−0.290 + 0.167i)47-s + (1.23 + 0.710i)53-s + 0.599·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ 0.875 - 0.483i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.444683908\)
\(L(\frac12)\) \(\approx\) \(2.444683908\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-3.08 + 1.77i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-8.03 - 4.63i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 4.24T + 169T^{2} \)
17 \( 1 + (-2.11 - 1.22i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-10.3 - 17.9i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-0.682 + 0.394i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 7.69iT - 841T^{2} \)
31 \( 1 + (-11.7 + 20.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-11.6 - 20.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 51.5iT - 1.68e3T^{2} \)
43 \( 1 + 49.3T + 1.84e3T^{2} \)
47 \( 1 + (13.6 - 7.88i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-65.1 - 37.6i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-13.6 - 7.88i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (4.01 + 6.94i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-43.6 + 75.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 40.0iT - 5.04e3T^{2} \)
73 \( 1 + (-17.2 + 29.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (4.34 + 7.52i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 115. iT - 6.88e3T^{2} \)
89 \( 1 + (81.6 - 47.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 154.T + 9.40e3T^{2} \)
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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

−9.404856342113000983287148979502, −8.370936172551473932620051494894, −7.70303859212487166512132652525, −6.64690137942777216214388366040, −5.99190950260470278062672484358, −5.15411237581294621635510636607, −4.20306017175854670408650515254, −3.27039005522938640080130515041, −1.96101497545133239488078605254, −1.09530232779165591545089677520, 0.74562766224195698735702903457, 1.98298108698039730137052035918, 3.04102127325449940462800514754, 3.94799831134153525153239712930, 5.07036033267342601458261098861, 5.87083584818589348444148070758, 6.67376831605301161233512312223, 7.28424705638786938159075518230, 8.477495252240691255925858408634, 8.976549704175781068923994896109

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