Properties

Label 2-21e2-441.130-c1-0-46
Degree $2$
Conductor $441$
Sign $-0.999 + 0.0210i$
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  + (0.442 − 0.301i)2-s + (−0.944 − 1.45i)3-s + (−0.625 + 1.59i)4-s + (−0.835 − 3.66i)5-s + (−0.855 − 0.357i)6-s + (1.43 − 2.22i)7-s + (0.442 + 1.93i)8-s + (−1.21 + 2.74i)9-s + (−1.47 − 1.36i)10-s + (−2.52 − 1.21i)11-s + (2.90 − 0.597i)12-s + (−2.75 + 1.87i)13-s + (−0.0356 − 1.41i)14-s + (−4.52 + 4.67i)15-s + (−1.73 − 1.60i)16-s + (−0.492 − 1.25i)17-s + ⋯
L(s)  = 1  + (0.312 − 0.213i)2-s + (−0.545 − 0.838i)3-s + (−0.312 + 0.797i)4-s + (−0.373 − 1.63i)5-s + (−0.349 − 0.145i)6-s + (0.542 − 0.840i)7-s + (0.156 + 0.685i)8-s + (−0.405 + 0.914i)9-s + (−0.466 − 0.432i)10-s + (−0.761 − 0.366i)11-s + (0.839 − 0.172i)12-s + (−0.763 + 0.520i)13-s + (−0.00951 − 0.378i)14-s + (−1.16 + 1.20i)15-s + (−0.432 − 0.401i)16-s + (−0.119 − 0.304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.999 + 0.0210i$
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: \((2,\ 441,\ (\ :1/2),\ -0.999 + 0.0210i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00726203 - 0.688299i\)
\(L(\frac12)\) \(\approx\) \(0.00726203 - 0.688299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.944 + 1.45i)T \)
7 \( 1 + (-1.43 + 2.22i)T \)
good2 \( 1 + (-0.442 + 0.301i)T + (0.730 - 1.86i)T^{2} \)
5 \( 1 + (0.835 + 3.66i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (2.52 + 1.21i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (2.75 - 1.87i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (0.492 + 1.25i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (-2.71 - 4.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.43 + 5.55i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (4.15 + 0.626i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (3.08 + 5.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.08 - 0.615i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (-0.669 - 0.206i)T + (33.8 + 23.0i)T^{2} \)
43 \( 1 + (4.39 - 1.35i)T + (35.5 - 24.2i)T^{2} \)
47 \( 1 + (-2.68 + 1.83i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-11.1 + 1.67i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-6.55 + 2.02i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (3.34 + 8.51i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-7.15 - 12.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.427 + 0.536i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.864 + 11.5i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-3.38 + 5.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.6 - 7.93i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (4.99 + 3.40i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (0.364 + 0.630i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.09241326103223220326111556761, −9.751501573482363269917659841481, −8.432700339834087576785658179583, −7.982703774247092511804070187828, −7.27882693526873556702152929517, −5.58791237308968729067346192480, −4.79921851019894179861208776932, −3.99395141636082155189526258308, −2.06345351907615337628387415943, −0.40616338950896333918212031643, 2.54044094759444409982676990295, 3.81046462560657087841928709399, 5.12189475296501730965680910793, 5.63216096100342577125813296591, 6.76094726909356668708939333937, 7.68235338128489102220253483139, 9.183900339396171279940115899626, 9.998348010621010737471026867386, 10.66661001860757898597675773124, 11.31287853543676489943357446897

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