Properties

Label 2-42e2-49.40-c0-0-0
Degree $2$
Conductor $1764$
Sign $0.886 - 0.462i$
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  + (0.733 + 0.680i)7-s + (−0.290 − 0.0663i)13-s + (1.17 + 0.680i)19-s + (−0.733 + 0.680i)25-s + (1.72 − 0.997i)31-s + (−0.123 + 1.64i)37-s + (−1.19 − 1.49i)43-s + (0.0747 + 0.997i)49-s + (0.865 + 0.0648i)61-s + (−0.365 − 0.632i)67-s + (0.766 + 0.825i)73-s + (−0.0747 + 0.129i)79-s + (−0.167 − 0.246i)91-s − 1.94i·97-s + (0.548 − 0.215i)103-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)7-s + (−0.290 − 0.0663i)13-s + (1.17 + 0.680i)19-s + (−0.733 + 0.680i)25-s + (1.72 − 0.997i)31-s + (−0.123 + 1.64i)37-s + (−1.19 − 1.49i)43-s + (0.0747 + 0.997i)49-s + (0.865 + 0.0648i)61-s + (−0.365 − 0.632i)67-s + (0.766 + 0.825i)73-s + (−0.0747 + 0.129i)79-s + (−0.167 − 0.246i)91-s − 1.94i·97-s + (0.548 − 0.215i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.886 - 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.234810305\)
\(L(\frac12)\) \(\approx\) \(1.234810305\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.733 - 0.680i)T \)
good5 \( 1 + (0.733 - 0.680i)T^{2} \)
11 \( 1 + (0.826 - 0.563i)T^{2} \)
13 \( 1 + (0.290 + 0.0663i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + (-1.17 - 0.680i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (-1.72 + 0.997i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.123 - 1.64i)T + (-0.988 - 0.149i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (1.19 + 1.49i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.0747 - 0.997i)T^{2} \)
53 \( 1 + (-0.988 + 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (-0.865 - 0.0648i)T + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.766 - 0.825i)T + (-0.0747 + 0.997i)T^{2} \)
79 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
97 \( 1 + 1.94iT - T^{2} \)
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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

−9.701580440950616233246559500343, −8.613257707707270488623227041730, −8.063337092514712810207827344128, −7.30201878511433081899063394356, −6.25924633511470897511475498376, −5.43432832683261127388011543105, −4.77455991568665451288862846589, −3.63339628918369463797939553629, −2.57817256658125249591440327156, −1.46106891732136234811089879572, 1.11219254727807093144761130914, 2.43204871924408024025778435751, 3.56497684144339151846282817276, 4.58475324359323394112303627777, 5.18132579270635553157223875424, 6.31222941964213960508474724466, 7.14746585351734644816208800174, 7.83809334844241547926134500708, 8.531948986353964913766185922140, 9.529229448341835332260341091734

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