Properties

Label 2-21e2-49.9-c1-0-10
Degree $2$
Conductor $441$
Sign $-0.351 + 0.936i$
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  + (−1.51 − 1.40i)2-s + (0.169 + 2.25i)4-s + (0.0830 + 0.211i)5-s + (1.07 − 2.41i)7-s + (0.338 − 0.424i)8-s + (0.171 − 0.436i)10-s + (4.81 − 1.48i)11-s + (−1.11 + 4.90i)13-s + (−5.02 + 2.14i)14-s + (3.36 − 0.507i)16-s + (−2.03 − 1.38i)17-s + (1.49 − 2.58i)19-s + (−0.463 + 0.223i)20-s + (−9.37 − 4.51i)22-s + (3.79 − 2.58i)23-s + ⋯
L(s)  = 1  + (−1.07 − 0.993i)2-s + (0.0845 + 1.12i)4-s + (0.0371 + 0.0946i)5-s + (0.407 − 0.913i)7-s + (0.119 − 0.150i)8-s + (0.0542 − 0.138i)10-s + (1.45 − 0.447i)11-s + (−0.310 + 1.36i)13-s + (−1.34 + 0.572i)14-s + (0.841 − 0.126i)16-s + (−0.493 − 0.336i)17-s + (0.342 − 0.593i)19-s + (−0.103 + 0.0499i)20-s + (−1.99 − 0.962i)22-s + (0.791 − 0.539i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.351 + 0.936i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.351 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479874 - 0.692801i\)
\(L(\frac12)\) \(\approx\) \(0.479874 - 0.692801i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.07 + 2.41i)T \)
good2 \( 1 + (1.51 + 1.40i)T + (0.149 + 1.99i)T^{2} \)
5 \( 1 + (-0.0830 - 0.211i)T + (-3.66 + 3.40i)T^{2} \)
11 \( 1 + (-4.81 + 1.48i)T + (9.08 - 6.19i)T^{2} \)
13 \( 1 + (1.11 - 4.90i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (2.03 + 1.38i)T + (6.21 + 15.8i)T^{2} \)
19 \( 1 + (-1.49 + 2.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.79 + 2.58i)T + (8.40 - 21.4i)T^{2} \)
29 \( 1 + (0.637 - 0.307i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (4.05 + 7.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.200 - 2.67i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (-1.10 + 1.38i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (0.910 + 1.14i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (4.11 + 3.81i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (-0.329 - 4.39i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (-2.06 + 5.26i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (-0.642 + 8.57i)T + (-60.3 - 9.09i)T^{2} \)
67 \( 1 + (-5.97 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.34 + 1.12i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (2.04 - 1.89i)T + (5.45 - 72.7i)T^{2} \)
79 \( 1 + (-2.40 + 4.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.53 - 6.73i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-9.81 - 3.02i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 - 0.497T + 97T^{2} \)
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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

−11.04030729536461974943677908461, −9.877248677073797544801568251362, −9.180444561526774944691562499763, −8.543546700711025336779351905248, −7.25928995874242580751270283277, −6.49921578498513951877438797022, −4.71489295080550718353331455968, −3.64331045904163412445564480012, −2.15272251730103728683823392625, −0.879677070960951300962396596148, 1.41905100250357820306475292076, 3.35261194610869583497469228886, 5.07293649328838668986673863175, 5.96486026604588480048041411583, 6.95860172804696422228812155509, 7.78488705159432614942585376629, 8.776973060669325797945895994141, 9.200554463623448634056392945192, 10.16567871265609252423415065497, 11.23466428902501187311298160266

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