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Properties

Label 2-42e2-28.23-c0-0-3
Degree $2$
Conductor $1764$
Sign $0.605 - 0.795i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.999·8-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)25-s + 2·29-s + (−0.499 − 0.866i)32-s + (1 − 1.73i)37-s − 0.999·50-s + (1 + 1.73i)53-s + (−1 + 1.73i)58-s + 0.999·64-s + (0.999 + 1.73i)74-s + (0.499 − 0.866i)100-s − 1.99·106-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.999·8-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)25-s + 2·29-s + (−0.499 − 0.866i)32-s + (1 − 1.73i)37-s − 0.999·50-s + (1 + 1.73i)53-s + (−1 + 1.73i)58-s + 0.999·64-s + (0.999 + 1.73i)74-s + (0.499 − 0.866i)100-s − 1.99·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8596814840\)
\(L(\frac12)\) \(\approx\) \(0.8596814840\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - 2T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - T^{2} \)
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

−9.337581459673687363659398076623, −8.843395094837522801935662493215, −7.961886229235568544866516932729, −7.29061222504411403034820649621, −6.49732187896102993517176655907, −5.72147085061260771859413813266, −4.87277509736667880743245682093, −3.99254377228003626569963720388, −2.57919268856991777045195546504, −1.10968720498954132696863325158, 1.04314858695646494344930395414, 2.39031932236456573789097631314, 3.19654849378673287674732862563, 4.31603067362702009851691691874, 4.99206306610946454868057275702, 6.32711052562153985071805056400, 7.09284282201505262503574200672, 8.257799438776743193375902765742, 8.449459910228350359406062402185, 9.570130037968544630607048077621

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