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Properties

Label 2-42e2-4.3-c0-0-0
Degree $2$
Conductor $1764$
Sign $1$
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 + i·8-s + 2i·11-s + 16-s + 2·22-s + 2i·23-s − 25-s − i·32-s + 2·37-s − 2i·44-s + 2·46-s + i·50-s − 64-s − 2i·71-s + ⋯
L(s)  = 1  − i·2-s − 4-s + i·8-s + 2i·11-s + 16-s + 2·22-s + 2i·23-s − 25-s − i·32-s + 2·37-s − 2i·44-s + 2·46-s + i·50-s − 64-s − 2i·71-s + ⋯

Functional equation

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

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: no
Arithmetic: yes
Character: $\chi_{1764} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9161266869\)
\(L(\frac12)\) \(\approx\) \(0.9161266869\)
\(L(1)\) 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 \( 1 + T^{2} \)
11 \( 1 - 2iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 2iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 2T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
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

−9.677534016422554380889696489099, −9.087226457401992645957959057159, −7.79851068674101803897170280456, −7.46584590131062865531149361679, −6.14094826546581014578261410632, −5.14495973646711114181274785782, −4.40399428588893855454513683357, −3.58397151224920566203758156222, −2.36410989565923920772025707143, −1.54863206611834761541176199307, 0.74598289977922072572841224702, 2.75162345285640079406163651692, 3.81010106382271513261700290318, 4.64474357843096484590921649091, 5.81092166993369817941610123598, 6.10961685892133376495046241716, 7.04563518481578072468631546470, 8.136044192715390128572271088127, 8.405083282613219135238751943892, 9.246431659450565247006921931598

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