Properties

Degree $2$
Conductor $441$
Sign $-0.420 - 0.907i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.335 + 0.580i)2-s + (1.27 + 1.17i)3-s + (0.775 + 1.34i)4-s + (0.712 + 1.23i)5-s + (−1.10 + 0.347i)6-s − 2.38·8-s + (0.252 + 2.98i)9-s − 0.955·10-s + (2.46 − 4.27i)11-s + (−0.585 + 2.62i)12-s + (1.37 + 2.38i)13-s + (−0.537 + 2.40i)15-s + (−0.752 + 1.30i)16-s + 1.11·17-s + (−1.82 − 0.855i)18-s − 4.01·19-s + ⋯
L(s)  = 1  + (−0.236 + 0.410i)2-s + (0.736 + 0.676i)3-s + (0.387 + 0.671i)4-s + (0.318 + 0.551i)5-s + (−0.452 + 0.141i)6-s − 0.841·8-s + (0.0843 + 0.996i)9-s − 0.302·10-s + (0.743 − 1.28i)11-s + (−0.168 + 0.756i)12-s + (0.381 + 0.661i)13-s + (−0.138 + 0.621i)15-s + (−0.188 + 0.326i)16-s + 0.271·17-s + (−0.429 − 0.201i)18-s − 0.921·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.420 - 0.907i$
Motivic weight: \(1\)
Character: $\chi_{441} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.420 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.957114 + 1.49763i\)
\(L(\frac12)\) \(\approx\) \(0.957114 + 1.49763i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.27 - 1.17i)T \)
7 \( 1 \)
good2 \( 1 + (0.335 - 0.580i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.712 - 1.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.46 + 4.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.37 - 2.38i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 + 4.01T + 19T^{2} \)
23 \( 1 + (2.71 + 4.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.40 + 5.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 + (0.124 + 0.215i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.498 - 0.863i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.73 - 8.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.820T + 53T^{2} \)
59 \( 1 + (3.29 + 5.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0376 + 0.0651i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.29 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.0804T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + (-0.922 + 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.23 + 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + (2.70 - 4.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.22417150214554430764673060613, −10.50195586372068537049899971304, −9.402550746192264098208300005982, −8.526074022974418139360773452198, −8.087172541614004511644388911966, −6.66371168991851616846009091697, −6.13359878899931523601819068927, −4.33865662694405425909123536113, −3.38059629205463086175306585614, −2.36928308147583675035593101476, 1.26335148749866537888469935672, 2.13595004979518976371711111142, 3.57791503827437694115003077393, 5.12614280858313368695622524933, 6.29987216846510139950965509902, 7.08450949924504172557039552917, 8.247528481550742489907738690407, 9.202314938671597813432573726230, 9.759233649288479062364279925893, 10.71895156646707031746508687621

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