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Properties

Label 2-42e2-84.47-c0-0-4
Degree $2$
Conductor $1764$
Sign $0.999 + 0.00394i$
Analytic cond. $0.880350$
Root an. cond. $0.938270$
Motivic weight $0$
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 + (0.499 + 0.866i)4-s + (0.382 − 0.662i)5-s − 0.999i·8-s + (−0.662 + 0.382i)10-s + 1.84i·13-s + (−0.5 + 0.866i)16-s + (0.923 + 1.60i)17-s + 0.765·20-s + (0.207 + 0.358i)25-s + (0.923 − 1.60i)26-s + (0.866 − 0.499i)32-s − 1.84i·34-s + (0.707 − 1.22i)37-s + (−0.662 − 0.382i)40-s − 1.84·41-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (0.382 − 0.662i)5-s − 0.999i·8-s + (−0.662 + 0.382i)10-s + 1.84i·13-s + (−0.5 + 0.866i)16-s + (0.923 + 1.60i)17-s + 0.765·20-s + (0.207 + 0.358i)25-s + (0.923 − 1.60i)26-s + (0.866 − 0.499i)32-s − 1.84i·34-s + (0.707 − 1.22i)37-s + (−0.662 − 0.382i)40-s − 1.84·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.00394i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.999 + 0.00394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8203306939\)
\(L(\frac12)\) \(\approx\) \(0.8203306939\)
\(L(1)\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.84iT - T^{2} \)
17 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.84T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 0.765iT - T^{2} \)
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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

−9.468780065209478078249104194928, −8.774284938377238803028031817141, −8.239429581485595079217241498317, −7.21071052449748517352230573887, −6.48799675609061171936964665971, −5.50601215422280481086706455548, −4.28679540427991386611786632376, −3.56293965515061761428463114419, −2.10166551084085750235790744150, −1.39456597428294982672270065276, 0.930240844354372354396917701343, 2.53411694799619878126008621084, 3.19947515459178824985509803432, 4.99228901949676871278130817293, 5.55702853901441104160140811979, 6.46086237627136702080673844101, 7.21038366472526322818030727045, 7.911374702681824068496355014977, 8.573081110102529831628874138354, 9.670194540165163926093693296922

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