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Properties

Label 2-21e2-49.25-c1-0-16
Degree $2$
Conductor $441$
Sign $0.948 + 0.316i$
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  + (2.06 − 0.638i)2-s + (2.21 − 1.51i)4-s + (2.32 + 0.351i)5-s + (0.990 + 2.45i)7-s + (0.926 − 1.16i)8-s + (5.04 − 0.760i)10-s + (−2.76 − 2.56i)11-s + (−0.814 + 3.56i)13-s + (3.61 + 4.44i)14-s + (−0.787 + 2.00i)16-s + (0.360 − 4.81i)17-s + (−1.58 − 2.73i)19-s + (5.70 − 2.74i)20-s + (−7.36 − 3.54i)22-s + (−0.496 − 6.61i)23-s + ⋯
L(s)  = 1  + (1.46 − 0.451i)2-s + (1.10 − 0.756i)4-s + (1.04 + 0.157i)5-s + (0.374 + 0.927i)7-s + (0.327 − 0.410i)8-s + (1.59 − 0.240i)10-s + (−0.834 − 0.774i)11-s + (−0.225 + 0.989i)13-s + (0.966 + 1.18i)14-s + (−0.196 + 0.501i)16-s + (0.0874 − 1.16i)17-s + (−0.362 − 0.628i)19-s + (1.27 − 0.614i)20-s + (−1.57 − 0.756i)22-s + (−0.103 − 1.38i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.948 + 0.316i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.25125 - 0.528684i\)
\(L(\frac12)\) \(\approx\) \(3.25125 - 0.528684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.990 - 2.45i)T \)
good2 \( 1 + (-2.06 + 0.638i)T + (1.65 - 1.12i)T^{2} \)
5 \( 1 + (-2.32 - 0.351i)T + (4.77 + 1.47i)T^{2} \)
11 \( 1 + (2.76 + 2.56i)T + (0.822 + 10.9i)T^{2} \)
13 \( 1 + (0.814 - 3.56i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.360 + 4.81i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (1.58 + 2.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.496 + 6.61i)T + (-22.7 + 3.42i)T^{2} \)
29 \( 1 + (5.84 - 2.81i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-1.61 + 2.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.09 - 2.10i)T + (13.5 + 34.4i)T^{2} \)
41 \( 1 + (-1.36 + 1.71i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (3.54 + 4.44i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-11.1 + 3.43i)T + (38.8 - 26.4i)T^{2} \)
53 \( 1 + (7.51 - 5.12i)T + (19.3 - 49.3i)T^{2} \)
59 \( 1 + (-5.46 + 0.823i)T + (56.3 - 17.3i)T^{2} \)
61 \( 1 + (-9.71 - 6.62i)T + (22.2 + 56.7i)T^{2} \)
67 \( 1 + (6.74 - 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.622 - 0.299i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (5.47 + 1.68i)T + (60.3 + 41.1i)T^{2} \)
79 \( 1 + (0.858 + 1.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.24 - 5.46i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (10.3 - 9.57i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + 2.26T + 97T^{2} \)
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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

−11.35572921652578397227829028333, −10.51601848169585036661522544878, −9.334053119329181876688252128014, −8.526977162760090556924640907688, −6.93382552631034611058201227432, −5.88932901073973849414348418154, −5.33432437850162016598587178178, −4.36171806456759072399832881318, −2.73504581199899674478920751273, −2.22517615046759918810461899587, 1.92608146008346027751734463961, 3.44687805092461953519106555280, 4.51130094686232947767014078616, 5.48468999751071861244099247538, 6.05010577641845495077901814355, 7.32572785353227002742287262811, 7.984649884641564698980385102008, 9.677851300564894739633478978134, 10.26142974540539167646508097750, 11.29184619214099544422173128038

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