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Properties

Label 2-42e2-7.5-c2-0-3
Degree $2$
Conductor $1764$
Sign $-0.937 - 0.347i$
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  + (0.804 + 0.464i)5-s + (4.84 + 8.39i)11-s + 15.9i·13-s + (−9.14 + 5.27i)17-s + (6.25 + 3.61i)19-s + (−5.65 + 9.78i)23-s + (−12.0 − 20.9i)25-s − 46.3·29-s + (−0.418 + 0.241i)31-s + (−1.24 + 2.14i)37-s − 55.8i·41-s + 60.6·43-s + (−31.6 − 18.2i)47-s + (−14.2 − 24.7i)53-s + 9.00i·55-s + ⋯
L(s)  = 1  + (0.160 + 0.0929i)5-s + (0.440 + 0.762i)11-s + 1.22i·13-s + (−0.537 + 0.310i)17-s + (0.329 + 0.190i)19-s + (−0.245 + 0.425i)23-s + (−0.482 − 0.836i)25-s − 1.59·29-s + (−0.0134 + 0.00779i)31-s + (−0.0335 + 0.0580i)37-s − 1.36i·41-s + 1.41·43-s + (−0.674 − 0.389i)47-s + (−0.269 − 0.466i)53-s + 0.163i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.937 - 0.347i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7864080148\)
\(L(\frac12)\) \(\approx\) \(0.7864080148\)
\(L(2)\) 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 \( 1 + (-0.804 - 0.464i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-4.84 - 8.39i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 15.9iT - 169T^{2} \)
17 \( 1 + (9.14 - 5.27i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.25 - 3.61i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (5.65 - 9.78i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 46.3T + 841T^{2} \)
31 \( 1 + (0.418 - 0.241i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (1.24 - 2.14i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 55.8iT - 1.68e3T^{2} \)
43 \( 1 - 60.6T + 1.84e3T^{2} \)
47 \( 1 + (31.6 + 18.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (14.2 + 24.7i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (81.4 - 47.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-95.4 - 55.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (41.0 + 71.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 127.T + 5.04e3T^{2} \)
73 \( 1 + (40.0 - 23.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (9.35 - 16.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 59.6iT - 6.88e3T^{2} \)
89 \( 1 + (-61.5 - 35.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 102. iT - 9.40e3T^{2} \)
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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

−9.345585148453213482098640668117, −8.912031199769515648682227853465, −7.75847965699394018651819802501, −7.09290585966223103567610975600, −6.31936336102821030977924463819, −5.47769009745491569390850247714, −4.34195032237619980548315531026, −3.81284493364709382104281311972, −2.33507072530499587367460280425, −1.57912237955459410205901264197, 0.19851495973999947666861870103, 1.44657742652865145528254605264, 2.77792752265059611471016308506, 3.59342739661026951130054385799, 4.66804534478456496188364008437, 5.65072271740905337881000913037, 6.16864896014812067331929050043, 7.31449724675880916741535183037, 7.923763615666127284867556411427, 8.854744495471514517581466646091

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