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Properties

Label 2-21e2-441.4-c1-0-21
Degree $2$
Conductor $441$
Sign $-0.424 - 0.905i$
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  + (−0.132 + 1.77i)2-s + (1.41 − 0.993i)3-s + (−1.14 − 0.172i)4-s + (−0.788 + 3.45i)5-s + (1.57 + 2.64i)6-s + (2.57 − 0.593i)7-s + (−0.332 + 1.45i)8-s + (1.02 − 2.81i)9-s + (−6.02 − 1.85i)10-s + (1.33 − 0.643i)11-s + (−1.79 + 0.892i)12-s + (−0.172 + 2.30i)13-s + (0.710 + 4.64i)14-s + (2.31 + 5.68i)15-s + (−4.75 − 1.46i)16-s + (0.421 − 0.0635i)17-s + ⋯
L(s)  = 1  + (−0.0939 + 1.25i)2-s + (0.819 − 0.573i)3-s + (−0.572 − 0.0863i)4-s + (−0.352 + 1.54i)5-s + (0.641 + 1.08i)6-s + (0.974 − 0.224i)7-s + (−0.117 + 0.515i)8-s + (0.342 − 0.939i)9-s + (−1.90 − 0.587i)10-s + (0.402 − 0.193i)11-s + (−0.518 + 0.257i)12-s + (−0.0479 + 0.640i)13-s + (0.189 + 1.24i)14-s + (0.597 + 1.46i)15-s + (−1.18 − 0.366i)16-s + (0.102 − 0.0154i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.424 - 0.905i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.424 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985109 + 1.54931i\)
\(L(\frac12)\) \(\approx\) \(0.985109 + 1.54931i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.41 + 0.993i)T \)
7 \( 1 + (-2.57 + 0.593i)T \)
good2 \( 1 + (0.132 - 1.77i)T + (-1.97 - 0.298i)T^{2} \)
5 \( 1 + (0.788 - 3.45i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-1.33 + 0.643i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.172 - 2.30i)T + (-12.8 - 1.93i)T^{2} \)
17 \( 1 + (-0.421 + 0.0635i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (2.91 + 5.04i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.15 + 1.44i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (3.05 - 7.77i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + (-3.82 - 6.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.11 - 2.84i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (1.79 + 1.66i)T + (3.06 + 40.8i)T^{2} \)
43 \( 1 + (-8.01 + 7.43i)T + (3.21 - 42.8i)T^{2} \)
47 \( 1 + (0.218 - 2.92i)T + (-46.4 - 7.00i)T^{2} \)
53 \( 1 + (3.69 + 9.42i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-1.66 + 1.54i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (5.36 - 0.808i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (2.30 + 3.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.09 + 11.4i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-4.11 + 2.80i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-2.85 + 4.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.721 + 9.62i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (0.179 + 2.39i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (2.09 + 3.63i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.27991222484477902487872627342, −10.65870352383506915424188501098, −9.095107026444296925982868658320, −8.425077324293323670273116433491, −7.44853835787047699767874195036, −6.94914388790835907977315865625, −6.37438880126133336726433384941, −4.77481826195311113771410485904, −3.35287821410948611174717878495, −2.12953191936103031483138119381, 1.26542959598590903508698292237, 2.38080028795565434825719465038, 3.96548315230947218472958748571, 4.43143137138608388102807227192, 5.67475739100769257225177237518, 7.78157592959132095125774520982, 8.304064712594029050833654742374, 9.256313472048283841611562957185, 9.837901388900990072189416977794, 10.89314618754217904288086262441

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