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Properties

Label 2-42e2-84.59-c0-0-8
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  + (0.258 − 0.965i)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 + (−0.999 − i)22-s + (−0.707 − 1.22i)23-s + (0.5 − 0.866i)25-s − 1.41i·29-s + (0.965 − 0.258i)32-s + 2i·43-s + (−1.22 + 0.707i)44-s + (−1.36 + 0.366i)46-s + (−0.707 − 0.707i)50-s + (−1.22 − 0.707i)53-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)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 + (−0.999 − i)22-s + (−0.707 − 1.22i)23-s + (0.5 − 0.866i)25-s − 1.41i·29-s + (0.965 − 0.258i)32-s + 2i·43-s + (−1.22 + 0.707i)44-s + (−1.36 + 0.366i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.114972233\)
\(L(\frac12)\) \(\approx\) \(1.114972233\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
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

−9.377124230451240160338569289432, −8.483525132410675848436489804199, −8.022940120291773869131828362860, −6.40614183515116399892389040275, −6.07036944814744043290356472993, −4.84819029447022482586852992499, −4.10393148289898009377031135101, −3.18973274159834233120614987943, −2.24050145656720541149907748375, −0.834678844511689710610614715460, 1.67278834907672215413076138761, 3.28702009548182394533229471380, 4.10092296601415750846695063242, 5.01289350689286962193115530776, 5.70825225701598322385765906763, 6.80681933209717227295141138249, 7.19276721689179413991230496520, 8.026730228272483717606957010608, 9.021529598647636537266141796625, 9.454776200705221084807764747374

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