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Properties

Label 2-42e2-4.3-c0-0-5
Degree $2$
Conductor $1764$
Sign $1$
Analytic cond. $0.880350$
Root an. cond. $0.938270$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 16-s − 25-s + 2·29-s + 32-s − 2·37-s − 50-s − 2·53-s + 2·58-s + 64-s − 2·74-s − 100-s − 2·106-s − 2·109-s − 2·113-s + 2·116-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s + 16-s − 25-s + 2·29-s + 32-s − 2·37-s − 50-s − 2·53-s + 2·58-s + 64-s − 2·74-s − 100-s − 2·106-s − 2·109-s − 2·113-s + 2·116-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1764} (883, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.164294269\)
\(L(\frac12)\) \(\approx\) \(2.164294269\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 + T )^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 + T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
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

−9.660097117877875751075970017828, −8.523385212984946399996742772565, −7.80412069296996575698543080058, −6.87343909023565811747888821654, −6.25655731337977056200033436977, −5.32250240130365142923749322102, −4.58860943332935353988836095425, −3.64641291316134394921926208693, −2.75428981141023316645793269301, −1.60128763308036182476724022177, 1.60128763308036182476724022177, 2.75428981141023316645793269301, 3.64641291316134394921926208693, 4.58860943332935353988836095425, 5.32250240130365142923749322102, 6.25655731337977056200033436977, 6.87343909023565811747888821654, 7.80412069296996575698543080058, 8.523385212984946399996742772565, 9.660097117877875751075970017828

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