Properties

Label 2-42e2-63.58-c1-0-16
Degree $2$
Conductor $1764$
Sign $0.580 - 0.814i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (1.5 − 0.866i)3-s + (1.5 + 2.59i)5-s + (1.5 − 2.59i)9-s + (−1.5 + 2.59i)11-s + (−0.5 + 0.866i)13-s + (4.5 + 2.59i)15-s + (3 + 5.19i)17-s + (−2 + 3.46i)19-s + (1.5 + 2.59i)23-s + (−2 + 3.46i)25-s − 5.19i·27-s + (−1.5 − 2.59i)29-s − 5·31-s + 5.19i·33-s + (−1 + 1.73i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.670 + 1.16i)5-s + (0.5 − 0.866i)9-s + (−0.452 + 0.783i)11-s + (−0.138 + 0.240i)13-s + (1.16 + 0.670i)15-s + (0.727 + 1.26i)17-s + (−0.458 + 0.794i)19-s + (0.312 + 0.541i)23-s + (−0.400 + 0.692i)25-s − 0.999i·27-s + (−0.278 − 0.482i)29-s − 0.898·31-s + 0.904i·33-s + (−0.164 + 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.580 - 0.814i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.529006614\)
\(L(\frac12)\) \(\approx\) \(2.529006614\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.5 - 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.444803248046898725972062443729, −8.612481923755314808691004583157, −7.62176417970067082748136768778, −7.24555731338226662547463188010, −6.28727758504548404321621562170, −5.66175014176923779218161155225, −4.16456234783465249932817469006, −3.34135657814592957674806550639, −2.34443467113015809225062503324, −1.66932601825441563141567650067, 0.851548885483323495055053457413, 2.25046092660961627731653106877, 3.10482898655835285615667472816, 4.23435876004542937315464188037, 5.19654731104344314686480371203, 5.51517353989944088224359349341, 6.96630415067636708266318504514, 7.78879172725979189256679975026, 8.717039056589033700261647410812, 9.006892466279282178175879640765

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