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Properties

Label 2-21e2-9.4-c1-0-31
Degree $2$
Conductor $441$
Sign $-0.420 + 0.907i$
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  + (−0.335 − 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: $-0.420 + 0.907i$
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: \((2,\ 441,\ (\ :1/2),\ -0.420 + 0.907i)\)

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

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

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