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Properties

Label 2-21e2-49.8-c1-0-12
Degree $2$
Conductor $441$
Sign $0.595 + 0.803i$
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  + (−0.159 − 0.200i)2-s + (0.430 − 1.88i)4-s + (1.50 − 0.726i)5-s + (2.63 + 0.216i)7-s + (−0.909 + 0.437i)8-s + (−0.387 − 0.186i)10-s + (2.04 + 2.56i)11-s + (1.69 + 2.12i)13-s + (−0.378 − 0.563i)14-s + (−3.25 − 1.56i)16-s + (−0.947 − 4.15i)17-s − 2.59·19-s + (−0.721 − 3.15i)20-s + (0.187 − 0.819i)22-s + (1.32 − 5.82i)23-s + ⋯
L(s)  = 1  + (−0.113 − 0.141i)2-s + (0.215 − 0.942i)4-s + (0.675 − 0.325i)5-s + (0.996 + 0.0819i)7-s + (−0.321 + 0.154i)8-s + (−0.122 − 0.0589i)10-s + (0.616 + 0.772i)11-s + (0.469 + 0.589i)13-s + (−0.101 − 0.150i)14-s + (−0.813 − 0.391i)16-s + (−0.229 − 1.00i)17-s − 0.595·19-s + (−0.161 − 0.706i)20-s + (0.0398 − 0.174i)22-s + (0.277 − 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.595 + 0.803i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.595 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50903 - 0.760397i\)
\(L(\frac12)\) \(\approx\) \(1.50903 - 0.760397i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.216i)T \)
good2 \( 1 + (0.159 + 0.200i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (-1.50 + 0.726i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-2.04 - 2.56i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.69 - 2.12i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.947 + 4.15i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + 2.59T + 19T^{2} \)
23 \( 1 + (-1.32 + 5.82i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.403 - 1.76i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 1.19T + 31T^{2} \)
37 \( 1 + (-0.755 - 3.31i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-5.05 + 2.43i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (9.10 + 4.38i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (0.656 + 0.823i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (2.01 - 8.82i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (10.7 + 5.18i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (1.71 + 7.49i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 + (3.66 - 16.0i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (4.59 - 5.76i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 0.219T + 79T^{2} \)
83 \( 1 + (10.0 - 12.6i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-10.7 + 13.5i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 5.31T + 97T^{2} \)
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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

−11.01793905160202946203085789274, −10.04992118915863526454444579788, −9.261707999567479864409747046242, −8.555536681347300507023497996124, −7.09269318233834088891579997808, −6.27798221066853358739467057820, −5.15949704222186736660058308202, −4.43595518680061121222851504556, −2.30034443111729673725886692110, −1.36192404063387251167958798133, 1.77842190616300638990350274413, 3.21450709404782154983340105201, 4.28944879112150927488551508266, 5.79526366690247253470800866849, 6.53816439271316012114168447004, 7.80263409326595651530333505310, 8.363156010118297458625428731536, 9.278836374800547106387892907727, 10.55092863597117331215820079161, 11.23405026749006623284773377990

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