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Properties

Label 2-21e2-441.4-c1-0-8
Degree $2$
Conductor $441$
Sign $-0.950 + 0.309i$
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  + (−0.192 + 2.56i)2-s + (1.55 − 0.766i)3-s + (−4.58 − 0.691i)4-s + (−0.144 + 0.633i)5-s + (1.66 + 4.13i)6-s + (−1.19 + 2.36i)7-s + (1.51 − 6.62i)8-s + (1.82 − 2.38i)9-s + (−1.60 − 0.493i)10-s + (−4.29 + 2.06i)11-s + (−7.65 + 2.43i)12-s + (−0.292 + 3.90i)13-s + (−5.83 − 3.52i)14-s + (0.260 + 1.09i)15-s + (7.86 + 2.42i)16-s + (−3.53 + 0.533i)17-s + ⋯
L(s)  = 1  + (−0.136 + 1.81i)2-s + (0.896 − 0.442i)3-s + (−2.29 − 0.345i)4-s + (−0.0647 + 0.283i)5-s + (0.681 + 1.68i)6-s + (−0.451 + 0.892i)7-s + (0.534 − 2.34i)8-s + (0.608 − 0.793i)9-s + (−0.506 − 0.156i)10-s + (−1.29 + 0.623i)11-s + (−2.20 + 0.704i)12-s + (−0.0810 + 1.08i)13-s + (−1.55 − 0.941i)14-s + (0.0673 + 0.282i)15-s + (1.96 + 0.606i)16-s + (−0.858 + 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.950 + 0.309i$
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: \((2,\ 441,\ (\ :1/2),\ -0.950 + 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175680 - 1.10749i\)
\(L(\frac12)\) \(\approx\) \(0.175680 - 1.10749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.55 + 0.766i)T \)
7 \( 1 + (1.19 - 2.36i)T \)
good2 \( 1 + (0.192 - 2.56i)T + (-1.97 - 0.298i)T^{2} \)
5 \( 1 + (0.144 - 0.633i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (4.29 - 2.06i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.292 - 3.90i)T + (-12.8 - 1.93i)T^{2} \)
17 \( 1 + (3.53 - 0.533i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.13 - 1.42i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-1.52 + 3.89i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + (1.48 + 2.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.55 - 6.51i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (-3.97 - 3.68i)T + (3.06 + 40.8i)T^{2} \)
43 \( 1 + (-9.26 + 8.60i)T + (3.21 - 42.8i)T^{2} \)
47 \( 1 + (-0.530 + 7.08i)T + (-46.4 - 7.00i)T^{2} \)
53 \( 1 + (-3.33 - 8.49i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-2.85 + 2.64i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (-7.83 + 1.18i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (-0.717 - 1.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.33 - 6.68i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-4.76 + 3.25i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (2.16 - 3.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.164 + 2.19i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (0.498 + 6.65i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (-8.43 - 14.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.93352086998520515553086961106, −10.14691396180482836090358957675, −9.346729965819300927380964434857, −8.658255801008896629480347894519, −7.80606757445863912535039972190, −7.09520940421562914827090923950, −6.29410701703333772156149410161, −5.26626278746899949768370930670, −4.01488046009320575071309585201, −2.39320358448028690555614571737, 0.66646100773755883378809255824, 2.58287978026015853651309968429, 3.17586028209969943228496387213, 4.31144882336757677998876127330, 5.18262848190704498618812553673, 7.35448457540136922519149818876, 8.361666581561046326325547529329, 9.084547789498590495112299710927, 9.959061098739079026511101536455, 10.78180453945010034166465245253

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