Properties

Label 2-21e2-9.4-c1-0-2
Degree $2$
Conductor $441$
Sign $0.998 - 0.0576i$
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  + (−0.849 − 1.47i)2-s + (−0.349 − 1.69i)3-s + (−0.444 + 0.769i)4-s + (−1.79 + 3.10i)5-s + (−2.19 + 1.95i)6-s − 1.88·8-s + (−2.75 + 1.18i)9-s + 6.09·10-s + (1.40 + 2.43i)11-s + (1.46 + 0.484i)12-s + (0.5 − 0.866i)13-s + (5.89 + 1.95i)15-s + (2.49 + 4.31i)16-s + 4.11·17-s + (4.08 + 3.04i)18-s + 0.888·19-s + ⋯
L(s)  = 1  + (−0.600 − 1.04i)2-s + (−0.201 − 0.979i)3-s + (−0.222 + 0.384i)4-s + (−0.802 + 1.38i)5-s + (−0.897 + 0.798i)6-s − 0.667·8-s + (−0.918 + 0.395i)9-s + 1.92·10-s + (0.423 + 0.733i)11-s + (0.421 + 0.139i)12-s + (0.138 − 0.240i)13-s + (1.52 + 0.505i)15-s + (0.623 + 1.07i)16-s + 0.997·17-s + (0.963 + 0.718i)18-s + 0.203·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.998 - 0.0576i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.998 - 0.0576i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560369 + 0.0161605i\)
\(L(\frac12)\) \(\approx\) \(0.560369 + 0.0161605i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.349 + 1.69i)T \)
7 \( 1 \)
good2 \( 1 + (0.849 + 1.47i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.79 - 3.10i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.40 - 2.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.11T + 17T^{2} \)
19 \( 1 - 0.888T + 19T^{2} \)
23 \( 1 + (2.93 - 5.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.849 - 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.49 - 6.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.76T + 37T^{2} \)
41 \( 1 + (2.70 - 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.60 + 4.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.33 + 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.123T + 53T^{2} \)
59 \( 1 + (4.43 - 7.68i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.93 - 3.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.15 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.87T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + (-3.54 - 6.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.05 + 3.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.60T + 89T^{2} \)
97 \( 1 + (-3.66 - 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.17001186335980134314590077303, −10.46575304990616899539240721570, −9.622109311090011893529297464922, −8.337338927098017137647627619448, −7.44938472982917005202134516285, −6.73902514002316537262089969391, −5.65192831644491885038353965160, −3.61833017210131173535566320256, −2.77189612292971444974006859678, −1.46006388277482293689985435444, 0.46588058026455834887683551563, 3.42609858146325835422102536206, 4.44304334442599282446269216354, 5.50197879626265285477364267750, 6.32515032313245620565985968892, 7.84015600172599961881578330912, 8.337403921812367076030571009938, 9.125482983379571244574968986799, 9.764531704227645092888679930000, 11.15567929584470547669552125102

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