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Properties

Label 2-21e2-49.13-c0-0-0
Degree $2$
Conductor $441$
Sign $0.999 + 0.0320i$
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  + (0.900 − 0.433i)4-s + (0.222 + 0.974i)7-s + (−1.52 + 0.347i)13-s + (0.623 − 0.781i)16-s − 1.94i·19-s + (−0.222 + 0.974i)25-s + (0.623 + 0.781i)28-s + 0.867i·31-s + (−1.62 − 0.781i)37-s + (0.277 − 0.347i)43-s + (−0.900 + 0.433i)49-s + (−1.22 + 0.974i)52-s + (−0.376 + 0.781i)61-s + (0.222 − 0.974i)64-s + 1.24·67-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)4-s + (0.222 + 0.974i)7-s + (−1.52 + 0.347i)13-s + (0.623 − 0.781i)16-s − 1.94i·19-s + (−0.222 + 0.974i)25-s + (0.623 + 0.781i)28-s + 0.867i·31-s + (−1.62 − 0.781i)37-s + (0.277 − 0.347i)43-s + (−0.900 + 0.433i)49-s + (−1.22 + 0.974i)52-s + (−0.376 + 0.781i)61-s + (0.222 − 0.974i)64-s + 1.24·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0320i$
Analytic conductor: \(0.220087\)
Root analytic conductor: \(0.469135\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :0),\ 0.999 + 0.0320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9913819414\)
\(L(\frac12)\) \(\approx\) \(0.9913819414\)
\(L(1)\) 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 \( 1 + (-0.222 - 0.974i)T \)
good2 \( 1 + (-0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (1.52 - 0.347i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 + 1.94iT - T^{2} \)
23 \( 1 + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (0.623 + 0.781i)T^{2} \)
31 \( 1 - 0.867iT - T^{2} \)
37 \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (0.376 - 0.781i)T + (-0.623 - 0.781i)T^{2} \)
67 \( 1 - 1.24T + T^{2} \)
71 \( 1 + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.846 + 0.193i)T + (0.900 + 0.433i)T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.900 + 0.433i)T^{2} \)
97 \( 1 - 1.94iT - T^{2} \)
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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

−11.42920279608250369262790393420, −10.54048706682459411598847510245, −9.504882862774435267936810518790, −8.793465225151556040295466600930, −7.37199799832499826984190017589, −6.84712120383781209023516684014, −5.55360292523801343958950518386, −4.86668972540395718839892426619, −2.93796922257296819669021008800, −2.01762822689431474824084583115, 1.91746541672123797073516136363, 3.29510686126521703266866764348, 4.41969686715835042120173225413, 5.78104499938859262269552110800, 6.88385193308016440535286434908, 7.65568389338397475151220458573, 8.273247674352147237500971157719, 9.987492537363328129202075676935, 10.28295572890717336624242386507, 11.41201889157146854518793628301

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