Properties

Label 2-21e2-63.16-c1-0-13
Degree $2$
Conductor $441$
Sign $-0.968 + 0.250i$
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  + (−1.26 − 2.19i)2-s + (−1.70 − 0.300i)3-s + (−2.20 + 3.82i)4-s + 0.879·5-s + (1.5 + 4.12i)6-s + 6.10·8-s + (2.81 + 1.02i)9-s + (−1.11 − 1.92i)10-s + 3.87·11-s + (4.91 − 5.85i)12-s + (−2.72 − 4.72i)13-s + (−1.49 − 0.264i)15-s + (−3.31 − 5.74i)16-s + (0.826 + 1.43i)17-s + (−1.31 − 7.48i)18-s + (1.20 − 2.08i)19-s + ⋯
L(s)  = 1  + (−0.895 − 1.55i)2-s + (−0.984 − 0.173i)3-s + (−1.10 + 1.91i)4-s + 0.393·5-s + (0.612 + 1.68i)6-s + 2.15·8-s + (0.939 + 0.342i)9-s + (−0.352 − 0.609i)10-s + 1.16·11-s + (1.41 − 1.68i)12-s + (−0.756 − 1.30i)13-s + (−0.387 − 0.0682i)15-s + (−0.829 − 1.43i)16-s + (0.200 + 0.347i)17-s + (−0.310 − 1.76i)18-s + (0.276 − 0.479i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0730635 - 0.573345i\)
\(L(\frac12)\) \(\approx\) \(0.0730635 - 0.573345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.300i)T \)
7 \( 1 \)
good2 \( 1 + (1.26 + 2.19i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.879T + 5T^{2} \)
11 \( 1 - 3.87T + 11T^{2} \)
13 \( 1 + (2.72 + 4.72i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.826 - 1.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.20 + 2.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.16T + 23T^{2} \)
29 \( 1 + (-3.02 + 5.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.27 - 3.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.27 + 3.94i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.592 + 1.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0923 - 0.160i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.511 + 0.885i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.64 + 6.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.33 + 5.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.29 + 2.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.47 + 2.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 + (6.39 + 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.97 - 5.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.109 - 0.189i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.51 + 9.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.25 + 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.73510132687580026383505222988, −9.956437572366119961624318949006, −9.400329304696706390481679477454, −8.208817212373027529675974570407, −7.19416373516975711698894193145, −5.90343120031207152084655717913, −4.65415282673417634060203467985, −3.36778734311101839185545425651, −1.93089288537058695160671753264, −0.64212283007056521566962026463, 1.33635310078923381695324664184, 4.27457840028778284698684265837, 5.24853283800690914433769717941, 6.19693244313385338104612223364, 6.82175618271114197410592741503, 7.56879030059443413393821810599, 8.990677774828053240270620311982, 9.502411266579388476361615818855, 10.20985835472778203779088554182, 11.44216917095541930114452751411

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