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Properties

Label 2-21e2-63.41-c1-0-2
Degree $2$
Conductor $441$
Sign $-0.178 - 0.983i$
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  + (0.105 − 0.0611i)2-s + (−0.792 − 1.54i)3-s + (−0.992 + 1.71i)4-s + (0.264 − 0.458i)5-s + (−0.178 − 0.114i)6-s + 0.487i·8-s + (−1.74 + 2.44i)9-s − 0.0647i·10-s + (−3.64 + 2.10i)11-s + (3.43 + 0.166i)12-s + (1.74 + 1.00i)13-s + (−0.915 − 0.0444i)15-s + (−1.95 − 3.38i)16-s − 4.38·17-s + (−0.0355 + 0.365i)18-s + 5.24i·19-s + ⋯
L(s)  = 1  + (0.0749 − 0.0432i)2-s + (−0.457 − 0.889i)3-s + (−0.496 + 0.859i)4-s + (0.118 − 0.205i)5-s + (−0.0727 − 0.0468i)6-s + 0.172i·8-s + (−0.581 + 0.813i)9-s − 0.0204i·10-s + (−1.09 + 0.633i)11-s + (0.991 + 0.0480i)12-s + (0.484 + 0.279i)13-s + (−0.236 − 0.0114i)15-s + (−0.488 − 0.846i)16-s − 1.06·17-s + (−0.00836 + 0.0861i)18-s + 1.20i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.178 - 0.983i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.178 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.387761 + 0.464574i\)
\(L(\frac12)\) \(\approx\) \(0.387761 + 0.464574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.792 + 1.54i)T \)
7 \( 1 \)
good2 \( 1 + (-0.105 + 0.0611i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.264 + 0.458i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.64 - 2.10i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.74 - 1.00i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.38T + 17T^{2} \)
19 \( 1 - 5.24iT - 19T^{2} \)
23 \( 1 + (-5.43 - 3.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.27 - 4.20i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.03 + 0.595i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.23T + 37T^{2} \)
41 \( 1 + (-0.0994 + 0.172i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.96 - 6.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.98 + 8.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.21iT - 53T^{2} \)
59 \( 1 + (-6.71 + 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.3 - 6.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.29 + 5.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.50iT - 71T^{2} \)
73 \( 1 - 5.61iT - 73T^{2} \)
79 \( 1 + (0.286 + 0.495i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.42 + 9.39i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + (0.493 - 0.285i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.44584698672861242958532218154, −10.76354336142456588537778394064, −9.380564075840275683707875796641, −8.519516838016244591551179901601, −7.61528464022445201147609089919, −6.96492343202531641489420704468, −5.58840561560599754014684840560, −4.75480215608379776907044419109, −3.30228219037483415220191616328, −1.85878419126486228878169511745, 0.38619506429697959919817434384, 2.75883211163795406319306481158, 4.24103457330641408595241159837, 5.11407666249290459851175584531, 5.88862548699536851107190577893, 6.85583712958787271470830509419, 8.536026976916518718280863633758, 9.124279206111483733105748316531, 10.12825275326960973688373057647, 10.96594326695503542468362200738

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