Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $-0.900 + 0.435i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.707 + 1.22i)5-s + 0.999·8-s + (−0.707 − 1.22i)10-s − 1.41·13-s + (−0.5 + 0.866i)16-s + (0.707 + 1.22i)17-s + 1.41·20-s + (−0.499 − 0.866i)25-s + (0.707 − 1.22i)26-s − 2·29-s + (−0.499 − 0.866i)32-s − 1.41·34-s + (−0.707 + 1.22i)40-s − 1.41·41-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.707 + 1.22i)5-s + 0.999·8-s + (−0.707 − 1.22i)10-s − 1.41·13-s + (−0.5 + 0.866i)16-s + (0.707 + 1.22i)17-s + 1.41·20-s + (−0.499 − 0.866i)25-s + (0.707 − 1.22i)26-s − 2·29-s + (−0.499 − 0.866i)32-s − 1.41·34-s + (−0.707 + 1.22i)40-s − 1.41·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.900 + 0.435i$
motivic weight  =  \(0\)
character  :  $\chi_{1764} (667, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :0),\ -0.900 + 0.435i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3533893065\)
\(L(\frac12)\)  \(\approx\)  \(0.3533893065\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.965702299377031751972798314830, −9.087465231311976125086648002661, −8.089576038383647906862720564081, −7.48347638676569615473871292913, −7.02815258369489530782367059805, −6.11190708894644019578999840838, −5.29401768510124845186256481067, −4.19436994871180207183784261692, −3.26916145146759369928842089433, −1.91104522520550825427703019289, 0.31003323001282393933179788417, 1.71010602305280053563800764311, 2.93786377108900925863387528571, 3.93631867236240465507806718378, 4.83802481538778707204668618847, 5.34344098036039832425231811150, 7.11447671973884288167442989895, 7.65654831200167262863158970697, 8.351754668431760920939286560386, 9.276259184125496606630298156791

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