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Properties

Label 2-21e2-49.22-c1-0-16
Degree $2$
Conductor $441$
Sign $-0.648 + 0.760i$
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  + (−1.72 + 0.832i)2-s + (1.05 − 1.31i)4-s + (0.379 − 1.66i)5-s + (0.548 − 2.58i)7-s + (0.134 − 0.589i)8-s + (0.728 + 3.19i)10-s + (−0.764 + 0.368i)11-s + (−4.30 + 2.07i)13-s + (1.20 + 4.93i)14-s + (1.00 + 4.41i)16-s + (−2.33 − 2.92i)17-s − 7.35·19-s + (−1.79 − 2.24i)20-s + (1.01 − 1.27i)22-s + (−3.45 + 4.33i)23-s + ⋯
L(s)  = 1  + (−1.22 + 0.588i)2-s + (0.525 − 0.658i)4-s + (0.169 − 0.743i)5-s + (0.207 − 0.978i)7-s + (0.0475 − 0.208i)8-s + (0.230 + 1.00i)10-s + (−0.230 + 0.111i)11-s + (−1.19 + 0.575i)13-s + (0.322 + 1.31i)14-s + (0.252 + 1.10i)16-s + (−0.565 − 0.709i)17-s − 1.68·19-s + (−0.400 − 0.502i)20-s + (0.216 − 0.271i)22-s + (−0.720 + 0.902i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.648 + 0.760i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.648 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106587 - 0.230978i\)
\(L(\frac12)\) \(\approx\) \(0.106587 - 0.230978i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.548 + 2.58i)T \)
good2 \( 1 + (1.72 - 0.832i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (-0.379 + 1.66i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (0.764 - 0.368i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (4.30 - 2.07i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (2.33 + 2.92i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + 7.35T + 19T^{2} \)
23 \( 1 + (3.45 - 4.33i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (1.10 + 1.38i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 3.51T + 31T^{2} \)
37 \( 1 + (4.51 + 5.65i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (1.53 - 6.72i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (1.44 + 6.32i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (2.34 - 1.13i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-6.09 + 7.63i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-0.338 - 1.48i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (5.74 + 7.20i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 - 8.27T + 67T^{2} \)
71 \( 1 + (-3.77 + 4.73i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-2.95 - 1.42i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + 6.53T + 79T^{2} \)
83 \( 1 + (10.0 + 4.83i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (7.65 + 3.68i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 - 8.74T + 97T^{2} \)
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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

−10.40531829034865217898757721993, −9.739275948619058491938518666402, −8.942903234963558085587955382335, −8.109019268139007035874732304951, −7.25703027436491700007421494743, −6.55147127742412446295164916232, −5.00626035177723609575197855375, −4.07200406487654820997827113121, −1.89322609090062472349317297900, −0.22366767947669388602343959725, 2.09661371113722464858060086797, 2.75799747406023117851811186318, 4.68574201437796768649706861599, 5.94719921256389567513496849700, 7.00811806146915022691221514038, 8.322497700043660406032661861840, 8.602822567837656460497881988971, 9.862805401434427143140428697784, 10.43335706320772159254509223431, 11.07700574037304169694966064843

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