Properties

Label 2-21e2-441.151-c1-0-51
Degree $2$
Conductor $441$
Sign $-0.0453 + 0.998i$
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  + (2.08 − 1.00i)2-s + (1.51 − 0.847i)3-s + (2.09 − 2.63i)4-s + (−1.46 − 1.36i)5-s + (2.29 − 3.28i)6-s + (−2.55 + 0.675i)7-s + (0.704 − 3.08i)8-s + (1.56 − 2.56i)9-s + (−4.42 − 1.36i)10-s + (2.74 + 1.86i)11-s + (0.938 − 5.75i)12-s + (3.06 + 2.08i)13-s + (−4.65 + 3.98i)14-s + (−3.36 − 0.811i)15-s + (−0.133 − 0.583i)16-s + (−5.17 + 0.779i)17-s + ⋯
L(s)  = 1  + (1.47 − 0.710i)2-s + (0.871 − 0.489i)3-s + (1.04 − 1.31i)4-s + (−0.655 − 0.608i)5-s + (0.938 − 1.34i)6-s + (−0.966 + 0.255i)7-s + (0.248 − 1.09i)8-s + (0.520 − 0.853i)9-s + (−1.40 − 0.431i)10-s + (0.826 + 0.563i)11-s + (0.270 − 1.66i)12-s + (0.850 + 0.579i)13-s + (−1.24 + 1.06i)14-s + (−0.869 − 0.209i)15-s + (−0.0333 − 0.145i)16-s + (−1.25 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0453 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0453 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0453 + 0.998i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.0453 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35353 - 2.46280i\)
\(L(\frac12)\) \(\approx\) \(2.35353 - 2.46280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.51 + 0.847i)T \)
7 \( 1 + (2.55 - 0.675i)T \)
good2 \( 1 + (-2.08 + 1.00i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (1.46 + 1.36i)T + (0.373 + 4.98i)T^{2} \)
11 \( 1 + (-2.74 - 1.86i)T + (4.01 + 10.2i)T^{2} \)
13 \( 1 + (-3.06 - 2.08i)T + (4.74 + 12.1i)T^{2} \)
17 \( 1 + (5.17 - 0.779i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-0.930 - 1.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.869 + 2.21i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (-4.97 + 0.749i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 + 1.71T + 31T^{2} \)
37 \( 1 + (3.33 - 8.50i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (-0.0854 + 0.0263i)T + (33.8 - 23.0i)T^{2} \)
43 \( 1 + (-4.12 - 1.27i)T + (35.5 + 24.2i)T^{2} \)
47 \( 1 + (8.72 - 4.20i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.956 - 2.43i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (2.95 + 12.9i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-4.18 - 5.24i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 7.10T + 67T^{2} \)
71 \( 1 + (-6.33 + 7.94i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (6.80 - 4.64i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + (1.31 - 0.895i)T + (30.3 - 77.2i)T^{2} \)
89 \( 1 + (1.11 + 14.9i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (0.0150 - 0.0261i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.40185304043742576087230838175, −10.08622078469667837616219066498, −9.034097148797855445735991666701, −8.306043795529290369279715869178, −6.73308291395731041540924429649, −6.25825071214665222172101592052, −4.52458650476013436119601259442, −3.94741858374569295177169430068, −2.93160667698366576935586807702, −1.63425536748124165920309194969, 2.90125740734474973057717062998, 3.61607494985392626504189907999, 4.22627243703783712210295710472, 5.63325536032287492294072956902, 6.71521558365250812439285313083, 7.27888445496921609795408425808, 8.473698155531142907212774614411, 9.406323581565630429881921207711, 10.67011052652207991012497052591, 11.44172581588031713447110558461

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