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Properties

Label 2-21e2-441.142-c1-0-24
Degree $2$
Conductor $441$
Sign $-0.485 - 0.874i$
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  + (−1.02 + 2.61i)2-s + (1.47 + 0.910i)3-s + (−4.29 − 3.98i)4-s + (1.33 − 0.640i)5-s + (−3.88 + 2.91i)6-s + (0.948 − 2.46i)7-s + (9.76 − 4.70i)8-s + (1.34 + 2.68i)9-s + (0.309 + 4.12i)10-s + (2.77 + 3.48i)11-s + (−2.70 − 9.79i)12-s + (1.57 − 4.02i)13-s + (5.47 + 5.00i)14-s + (2.54 + 0.266i)15-s + (1.39 + 18.6i)16-s + (−1.37 + 1.27i)17-s + ⋯
L(s)  = 1  + (−0.724 + 1.84i)2-s + (0.850 + 0.525i)3-s + (−2.14 − 1.99i)4-s + (0.594 − 0.286i)5-s + (−1.58 + 1.18i)6-s + (0.358 − 0.933i)7-s + (3.45 − 1.66i)8-s + (0.447 + 0.894i)9-s + (0.0978 + 1.30i)10-s + (0.838 + 1.05i)11-s + (−0.780 − 2.82i)12-s + (0.437 − 1.11i)13-s + (1.46 + 1.33i)14-s + (0.656 + 0.0688i)15-s + (0.348 + 4.65i)16-s + (−0.333 + 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.485 - 0.874i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.485 - 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692312 + 1.17665i\)
\(L(\frac12)\) \(\approx\) \(0.692312 + 1.17665i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.47 - 0.910i)T \)
7 \( 1 + (-0.948 + 2.46i)T \)
good2 \( 1 + (1.02 - 2.61i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-1.33 + 0.640i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-2.77 - 3.48i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.57 + 4.02i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (1.37 - 1.27i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.601 - 1.04i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.20 + 5.29i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.85 - 0.571i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (0.861 - 1.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.63 - 1.43i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-0.243 - 0.166i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-3.85 + 2.62i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-1.75 + 4.46i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-0.863 + 0.266i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (8.97 - 6.12i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (6.96 - 6.46i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (0.615 - 1.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.485 - 2.12i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (11.5 + 1.74i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (6.42 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.22 - 8.20i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-1.90 - 4.85i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-4.61 + 7.99i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.57681327297680466686894174874, −10.20598106552895263636561128321, −9.283727063875839588694042177751, −8.630255185014700436326233636553, −7.76196817109856917951141359730, −7.05431244826318621377422976247, −5.95601302929402734171018261564, −4.79616585463588055087484538320, −4.09759327356059656094655640333, −1.41799831197561569547848296664, 1.38997200224983504165369272160, 2.30747769516625853265834159398, 3.25917860724575552360973300547, 4.36612078236221283399677832116, 6.17358327618889014972845866519, 7.66402350840822189634324309921, 8.655627324133698011240933547692, 9.158742028989702583640516446622, 9.662241447593421597238702236211, 11.06459489766089558085222861992

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