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Properties

Label 2-21e2-21.2-c0-0-1
Degree $2$
Conductor $441$
Sign $0.675 - 0.736i$
Analytic cond. $0.220087$
Root an. cond. $0.469135$
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  + (1.22 + 0.707i)2-s + (0.499 + 0.866i)4-s + (−1.22 + 0.707i)11-s + (0.499 − 0.866i)16-s − 2·22-s + (−1.22 − 0.707i)23-s + (−0.5 − 0.866i)25-s + 1.41i·29-s + (1.22 − 0.707i)32-s + (−1.22 − 0.707i)44-s + (−0.999 − 1.73i)46-s − 1.41i·50-s + (1.22 − 0.707i)53-s + (−1.00 + 1.73i)58-s + 0.999·64-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)2-s + (0.499 + 0.866i)4-s + (−1.22 + 0.707i)11-s + (0.499 − 0.866i)16-s − 2·22-s + (−1.22 − 0.707i)23-s + (−0.5 − 0.866i)25-s + 1.41i·29-s + (1.22 − 0.707i)32-s + (−1.22 − 0.707i)44-s + (−0.999 − 1.73i)46-s − 1.41i·50-s + (1.22 − 0.707i)53-s + (−1.00 + 1.73i)58-s + 0.999·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.675 - 0.736i$
Analytic conductor: \(0.220087\)
Root analytic conductor: \(0.469135\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :0),\ 0.675 - 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.441505518\)
\(L(\frac12)\) \(\approx\) \(1.441505518\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−11.84052208846762375279621408714, −10.49103574622163080226566462596, −9.864992801402636931402749329696, −8.427010539387864622455064897460, −7.51647339343988795042853942915, −6.65004984613259944318897430329, −5.63783728162440250647776145816, −4.83744785579779234334522478818, −3.86931961490569081585999189777, −2.47891726023275405362271729690, 2.14017190261486800664524466824, 3.25394871834383751516281348352, 4.24701759009354368989110402291, 5.41790831006616825823780459578, 5.99673600071139469592678931300, 7.59703851389044336707338211636, 8.385762160058102779869942834926, 9.750816994845927112963944443200, 10.63406120038908678855810746401, 11.43467133037809501503769290651

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