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Properties

Label 2-42e2-7.3-c2-0-20
Degree $2$
Conductor $1764$
Sign $0.982 + 0.188i$
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  + (−0.416 + 0.240i)5-s + (2.88 − 4.99i)11-s − 1.41i·13-s + (19.9 + 11.5i)17-s + (9.24 − 5.33i)19-s + (−3.29 − 5.71i)23-s + (−12.3 + 21.4i)25-s + 6.20·29-s + (−36.3 − 20.9i)31-s + (30.0 + 52.0i)37-s − 48.8i·41-s − 51.5·43-s + (16.6 − 9.62i)47-s + (41.0 − 71.1i)53-s + 2.77i·55-s + ⋯
L(s)  = 1  + (−0.0832 + 0.0480i)5-s + (0.261 − 0.453i)11-s − 0.109i·13-s + (1.17 + 0.678i)17-s + (0.486 − 0.280i)19-s + (−0.143 − 0.248i)23-s + (−0.495 + 0.858i)25-s + 0.213·29-s + (−1.17 − 0.676i)31-s + (0.812 + 1.40i)37-s − 1.19i·41-s − 1.19·43-s + (0.354 − 0.204i)47-s + (0.774 − 1.34i)53-s + 0.0503i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.982 + 0.188i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ 0.982 + 0.188i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.019686155\)
\(L(\frac12)\) \(\approx\) \(2.019686155\)
\(L(2)\) 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 \( 1 + (0.416 - 0.240i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.88 + 4.99i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 1.41iT - 169T^{2} \)
17 \( 1 + (-19.9 - 11.5i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.24 + 5.33i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (3.29 + 5.71i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 6.20T + 841T^{2} \)
31 \( 1 + (36.3 + 20.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-30.0 - 52.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 48.8iT - 1.68e3T^{2} \)
43 \( 1 + 51.5T + 1.84e3T^{2} \)
47 \( 1 + (-16.6 + 9.62i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-41.0 + 71.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-80.1 - 46.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-4.32 + 2.49i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.10 + 1.91i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 80.5T + 5.04e3T^{2} \)
73 \( 1 + (-12.0 - 6.95i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-32.4 - 56.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 118. iT - 6.88e3T^{2} \)
89 \( 1 + (-90.2 + 52.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 31.7iT - 9.40e3T^{2} \)
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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

−9.073806970389173944560121469757, −8.263514110890544745913172457346, −7.58001241953730028109389269124, −6.71902922444387114892708633314, −5.77967754491328997321386561039, −5.16208271295150915711066259385, −3.88629839869279001422306877572, −3.28274256116156936556329216173, −1.96611724444540968159689209820, −0.75256839401589152096288609196, 0.839072205205657334634412721737, 2.06035506472599342981329837082, 3.23092761865294386225873804752, 4.09990545435017366520938564810, 5.11214519204189161213239083523, 5.83777786750128384503678920280, 6.84872863709508935940298292632, 7.58116176177180902294929797480, 8.244887122444758828170900313016, 9.318521049342390099469268015100

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