Properties

Label 2-42e2-49.47-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  + (−0.955 − 0.294i)7-s + (1.81 − 0.414i)13-s + (−0.510 − 0.294i)19-s + (0.955 − 0.294i)25-s + (0.975 − 0.563i)31-s + (−0.123 + 0.0841i)37-s + (0.914 − 1.14i)43-s + (0.826 + 0.563i)49-s + (−0.488 − 0.716i)61-s + (0.988 + 1.71i)67-s + (−0.587 − 1.90i)73-s + (−0.826 + 1.43i)79-s + (−1.85 − 0.139i)91-s + 1.94i·97-s + (−0.202 − 1.34i)103-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)7-s + (1.81 − 0.414i)13-s + (−0.510 − 0.294i)19-s + (0.955 − 0.294i)25-s + (0.975 − 0.563i)31-s + (−0.123 + 0.0841i)37-s + (0.914 − 1.14i)43-s + (0.826 + 0.563i)49-s + (−0.488 − 0.716i)61-s + (0.988 + 1.71i)67-s + (−0.587 − 1.90i)73-s + (−0.826 + 1.43i)79-s + (−1.85 − 0.139i)91-s + 1.94i·97-s + (−0.202 − 1.34i)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: $\chi_{1764} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.886 + 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.089597769\)
\(L(\frac12)\) \(\approx\) \(1.089597769\)
\(L(1)\) 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 \( 1 + (0.955 + 0.294i)T \)
good5 \( 1 + (-0.955 + 0.294i)T^{2} \)
11 \( 1 + (0.0747 - 0.997i)T^{2} \)
13 \( 1 + (-1.81 + 0.414i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (0.988 - 0.149i)T^{2} \)
19 \( 1 + (0.510 + 0.294i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.988 - 0.149i)T^{2} \)
29 \( 1 + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.975 + 0.563i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.123 - 0.0841i)T + (0.365 - 0.930i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.914 + 1.14i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.826 - 0.563i)T^{2} \)
53 \( 1 + (0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 - 0.294i)T^{2} \)
61 \( 1 + (0.488 + 0.716i)T + (-0.365 + 0.930i)T^{2} \)
67 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.587 + 1.90i)T + (-0.826 + 0.563i)T^{2} \)
79 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.0747 - 0.997i)T^{2} \)
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

−9.330750922646498099133373597261, −8.692275670586855025054433759086, −7.957970859915843833448638079092, −6.85293260577674644039595012225, −6.32200678918440947954269025908, −5.53472141665947407518960640543, −4.28733769170551877340445281252, −3.55000175807754458354544858199, −2.59858640958885467647128549044, −0.993839710027635202613149549972, 1.32402336706378373783659359082, 2.76935023174560352958547014707, 3.60825410266652176689266513474, 4.50096995834584605858117878017, 5.73965726963788637150148330585, 6.33176769038514206868756224919, 6.97575889865620406736845733898, 8.170237068506724698088807464966, 8.792034041245095274606327704949, 9.423282713563092939196015140071

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