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Properties

Label 2-21e2-9.7-c1-0-4
Degree $2$
Conductor $441$
Sign $-0.958 + 0.285i$
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  + (−0.649 + 1.12i)2-s + (−0.0514 + 1.73i)3-s + (0.155 + 0.268i)4-s + (1.76 + 3.05i)5-s + (−1.91 − 1.18i)6-s − 3.00·8-s + (−2.99 − 0.177i)9-s − 4.58·10-s + (−0.589 + 1.02i)11-s + (−0.473 + 0.254i)12-s + (−1.61 − 2.78i)13-s + (−5.37 + 2.89i)15-s + (1.64 − 2.84i)16-s + 4.90·17-s + (2.14 − 3.25i)18-s + 6.86·19-s + ⋯
L(s)  = 1  + (−0.459 + 0.796i)2-s + (−0.0296 + 0.999i)3-s + (0.0775 + 0.134i)4-s + (0.788 + 1.36i)5-s + (−0.782 − 0.482i)6-s − 1.06·8-s + (−0.998 − 0.0593i)9-s − 1.44·10-s + (−0.177 + 0.307i)11-s + (−0.136 + 0.0735i)12-s + (−0.446 − 0.773i)13-s + (−1.38 + 0.747i)15-s + (0.410 − 0.710i)16-s + 1.18·17-s + (0.505 − 0.767i)18-s + 1.57·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.958 + 0.285i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169409 - 1.16135i\)
\(L(\frac12)\) \(\approx\) \(0.169409 - 1.16135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0514 - 1.73i)T \)
7 \( 1 \)
good2 \( 1 + (0.649 - 1.12i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.76 - 3.05i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.589 - 1.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.61 + 2.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.90T + 17T^{2} \)
19 \( 1 - 6.86T + 19T^{2} \)
23 \( 1 + (-2.14 - 3.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.36 + 2.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.960 + 1.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.76T + 37T^{2} \)
41 \( 1 + (-3.32 - 5.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 + 8.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.316 + 0.548i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.22T + 53T^{2} \)
59 \( 1 + (-4.10 - 7.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.82 - 8.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.66 + 4.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 + 1.03T + 73T^{2} \)
79 \( 1 + (0.502 - 0.869i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.65 + 6.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (-5.46 + 9.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.42288365087652961478760468276, −10.32436363902441507214215356120, −9.889075458900270768490726919055, −9.044500793386598342259378222191, −7.71926056447828282769617924430, −7.18147233145909976845486498768, −5.90271911765023812794348854592, −5.37195848341128253207902498382, −3.40674425552176473779701799699, −2.79043396785401570656660803024, 0.901736858071418795109087708504, 1.74591224187553181760597415472, 3.03442383935912762302656171262, 5.13127406102087744800810033066, 5.74420758648306475162073904161, 6.91033379447508933690849498215, 8.127997226775250509660455343507, 9.069908310792778171693505004338, 9.560049778460297622128504507368, 10.64347639414494620131993017833

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