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Properties

Label 2-21e2-441.130-c1-0-22
Degree $2$
Conductor $441$
Sign $0.997 + 0.0675i$
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  + (0.821 − 0.560i)2-s + (−1.08 − 1.35i)3-s + (−0.369 + 0.941i)4-s + (−0.0503 − 0.220i)5-s + (−1.64 − 0.507i)6-s + (1.99 + 1.73i)7-s + (0.666 + 2.91i)8-s + (−0.665 + 2.92i)9-s + (−0.164 − 0.152i)10-s + (2.39 + 1.15i)11-s + (1.67 − 0.517i)12-s + (2.29 − 1.56i)13-s + (2.61 + 0.304i)14-s + (−0.244 + 0.306i)15-s + (0.698 + 0.648i)16-s + (1.07 + 2.72i)17-s + ⋯
L(s)  = 1  + (0.580 − 0.396i)2-s + (−0.623 − 0.781i)3-s + (−0.184 + 0.470i)4-s + (−0.0225 − 0.0986i)5-s + (−0.671 − 0.206i)6-s + (0.755 + 0.655i)7-s + (0.235 + 1.03i)8-s + (−0.221 + 0.975i)9-s + (−0.0521 − 0.0483i)10-s + (0.723 + 0.348i)11-s + (0.483 − 0.149i)12-s + (0.635 − 0.433i)13-s + (0.698 + 0.0813i)14-s + (−0.0630 + 0.0790i)15-s + (0.174 + 0.162i)16-s + (0.259 + 0.661i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.997 + 0.0675i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.997 + 0.0675i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61368 - 0.0545483i\)
\(L(\frac12)\) \(\approx\) \(1.61368 - 0.0545483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08 + 1.35i)T \)
7 \( 1 + (-1.99 - 1.73i)T \)
good2 \( 1 + (-0.821 + 0.560i)T + (0.730 - 1.86i)T^{2} \)
5 \( 1 + (0.0503 + 0.220i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (-2.39 - 1.15i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-2.29 + 1.56i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (-1.07 - 2.72i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (3.31 + 5.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.25 - 4.08i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (0.700 + 0.105i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (-1.46 - 2.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.10 + 1.22i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (-11.2 - 3.47i)T + (33.8 + 23.0i)T^{2} \)
43 \( 1 + (-1.58 + 0.490i)T + (35.5 - 24.2i)T^{2} \)
47 \( 1 + (6.76 - 4.61i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-1.80 + 0.271i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (5.56 - 1.71i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (1.77 + 4.51i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-0.681 - 1.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.06 + 1.32i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.771 + 10.2i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-0.398 + 0.690i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.14 + 4.86i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (-1.36 - 0.929i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (-4.23 - 7.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.20888136520257611338113460539, −10.85506738399241869947191508945, −9.030341749431987596206717711313, −8.367272450868787291386423154065, −7.42367357250762198507412352767, −6.29520704421336494397507960519, −5.22493318973657481170972985596, −4.42879062650194505736566323879, −2.89617912357696462425872983069, −1.58659214188030921963302229811, 1.11283459568706269742338626779, 3.69689294033379597933768634859, 4.39355214899048217635026515509, 5.31879979130409897626584206492, 6.26104707430925677373967771421, 7.03579598680049040331764467054, 8.547755683523524379900302922593, 9.437121592378580078438494452102, 10.47560895365010007893346102460, 10.93539649970114243519938115790

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