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Properties

Label 2-42e2-84.83-c0-0-3
Degree $2$
Conductor $1764$
Sign $0.716 + 0.698i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.716 + 0.698i$
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: \((2,\ 1764,\ (\ :0),\ 0.716 + 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.198997622\)
\(L(\frac12)\) \(\approx\) \(1.198997622\)
\(L(1)\) 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

−9.628001262722872473059309256848, −8.914185426021120442912112640840, −8.025248531824381133577420633695, −7.02632590247201524920917971935, −5.95868994977902670022733959138, −5.24864124594555686740093600968, −4.24795734900663932042724248736, −3.40488846095353589649218946965, −2.22472491269030609607377519732, −1.42032917235380558682707499699, 1.11350700702207569549548662926, 2.87467717383615755648803791688, 3.79051453753473926564416174066, 5.09662785367335967692893747719, 5.66702740947841132006920157471, 6.14222939154688672314160188700, 7.43955392502499907967682555885, 7.78606491713818434352256599561, 8.677190779699083676792159151011, 9.557446086959149255098683077100

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