Properties

Label 2-42e2-84.83-c0-0-0
Degree $2$
Conductor $1764$
Sign $-0.896 + 0.442i$
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  + i·2-s − 4-s − 0.765·5-s i·8-s − 0.765i·10-s + 1.84i·13-s + 16-s − 1.84·17-s + 0.765·20-s − 0.414·25-s − 1.84·26-s + i·32-s − 1.84i·34-s − 1.41·37-s + 0.765i·40-s − 1.84·41-s + ⋯
L(s)  = 1  + i·2-s − 4-s − 0.765·5-s i·8-s − 0.765i·10-s + 1.84i·13-s + 16-s − 1.84·17-s + 0.765·20-s − 0.414·25-s − 1.84·26-s + i·32-s − 1.84i·34-s − 1.41·37-s + 0.765i·40-s − 1.84·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.896 + 0.442i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ -0.896 + 0.442i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3628726676\)
\(L(\frac12)\) \(\approx\) \(0.3628726676\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.765T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.84iT - T^{2} \)
17 \( 1 + 1.84T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 + 1.84T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 0.765iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 0.765iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.765T + T^{2} \)
97 \( 1 + 0.765iT - T^{2} \)
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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

−9.614227038473247761963582378482, −8.861690191637752517288200976965, −8.470874415411871964847949793160, −7.37286369535516147596222139474, −6.83554774700104499351194627973, −6.21164277243946646649480727585, −4.93481455176571975384976323156, −4.33909734910661958073156051152, −3.59292354269028910237545565392, −1.91706041128369932223437314601, 0.26632204724506199773764490625, 1.94654061436474676738336277594, 3.08092458944129551683799876104, 3.78761371303869051775689980247, 4.76634290584006806791108200997, 5.49227512797760681584148060602, 6.69297214205565871997125328914, 7.74916555843621085456337787344, 8.403656335657013308448349282119, 8.985573759870906334660058223607

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