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Properties

Label 2-21e2-21.20-c1-0-6
Degree $2$
Conductor $441$
Sign $0.716 + 0.698i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41i·2-s − 3.82·4-s − 3.37·5-s − 4.41i·8-s − 8.15i·10-s + 0.828i·11-s − 3.37i·13-s + 2.99·16-s − 1.39·17-s − 6.75i·19-s + 12.9·20-s − 1.99·22-s + 2i·23-s + 6.41·25-s + 8.15·26-s + ⋯
L(s)  = 1  + 1.70i·2-s − 1.91·4-s − 1.51·5-s − 1.56i·8-s − 2.57i·10-s + 0.249i·11-s − 0.937i·13-s + 0.749·16-s − 0.339·17-s − 1.55i·19-s + 2.89·20-s − 0.426·22-s + 0.417i·23-s + 1.28·25-s + 1.59·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.716 + 0.698i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.716 + 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124213 - 0.0505284i\)
\(L(\frac12)\) \(\approx\) \(0.124213 - 0.0505284i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41iT - 2T^{2} \)
5 \( 1 + 3.37T + 5T^{2} \)
11 \( 1 - 0.828iT - 11T^{2} \)
13 \( 1 + 3.37iT - 13T^{2} \)
17 \( 1 + 1.39T + 17T^{2} \)
19 \( 1 + 6.75iT - 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 + 4.82iT - 29T^{2} \)
31 \( 1 - 6.75iT - 31T^{2} \)
37 \( 1 + 2.58T + 37T^{2} \)
41 \( 1 + 8.15T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + 6.75T + 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + 6.75T + 59T^{2} \)
61 \( 1 - 8.15iT - 61T^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 - 4.82iT - 71T^{2} \)
73 \( 1 - 1.39iT - 73T^{2} \)
79 \( 1 + 9.65T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + 6.17T + 89T^{2} \)
97 \( 1 + 1.39iT - 97T^{2} \)
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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

−11.10110406867335002007065103745, −9.824579352893787174121630007490, −8.581658080247910116985946897593, −8.192457691599951078200763082437, −7.20559347526821162567416821407, −6.69887341838425057245998098021, −5.26180017069869164176242330026, −4.56081943841721085116607636033, −3.33378652358355589080491716145, −0.085685749917432787358306129667, 1.71650265971139107293354797900, 3.29971236218686794797978573628, 3.94954358646271740796109979890, 4.87804183958878609864324921893, 6.64960883111342865367819831198, 7.957529224980737383414260798664, 8.666581281075428059786470504643, 9.695590108917904403908026138380, 10.61854758582824296244164080603, 11.38133390386490500898106025361

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