Properties

Label 2-42e2-84.59-c0-0-3
Degree $2$
Conductor $1764$
Sign $0.627 - 0.778i$
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  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 + 0.707i)8-s + (−0.707 + 1.22i)11-s + (0.500 + 0.866i)16-s + (−1 + 0.999i)22-s + (0.707 + 1.22i)23-s + (0.5 − 0.866i)25-s − 1.41i·29-s + (0.258 + 0.965i)32-s − 2i·43-s + (−1.22 + 0.707i)44-s + (0.366 + 1.36i)46-s + (0.707 − 0.707i)50-s + (−1.22 − 0.707i)53-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 + 0.707i)8-s + (−0.707 + 1.22i)11-s + (0.500 + 0.866i)16-s + (−1 + 0.999i)22-s + (0.707 + 1.22i)23-s + (0.5 − 0.866i)25-s − 1.41i·29-s + (0.258 + 0.965i)32-s − 2i·43-s + (−1.22 + 0.707i)44-s + (0.366 + 1.36i)46-s + (0.707 − 0.707i)50-s + (−1.22 − 0.707i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.627 - 0.778i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.627 - 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.028612162\)
\(L(\frac12)\) \(\approx\) \(2.028612162\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 2iT - T^{2} \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.748964978949175800518305992141, −8.645498604320676413142903174710, −7.71191801392287446066436844166, −7.21503397241020951343478397556, −6.34435916528212467240360821044, −5.40975860610207826418637053202, −4.74637655061419750970090766859, −3.89454210711016425804671586639, −2.79765304332580955171098204528, −1.88786330712882445247689969452, 1.28190411002428933982519172968, 2.79441453195413899931943386968, 3.26795231523470101728585805068, 4.51679156479561831040157366173, 5.21370321704005017961418917439, 6.03395634063371128019131976239, 6.78208831686678674393145666608, 7.67732153791640869113703597527, 8.570698638588248276316216256421, 9.428799761603366880821295683230

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