Properties

Label 2-21e2-63.4-c1-0-9
Degree $2$
Conductor $441$
Sign $0.940 - 0.339i$
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  + (−1.23 + 2.13i)2-s + (−0.410 + 1.68i)3-s + (−2.02 − 3.51i)4-s − 3.65·5-s + (−3.08 − 2.94i)6-s + 5.05·8-s + (−2.66 − 1.38i)9-s + (4.50 − 7.79i)10-s + 0.406·11-s + (6.73 − 1.97i)12-s + (0.243 − 0.421i)13-s + (1.5 − 6.15i)15-s + (−2.16 + 3.74i)16-s + (−2.42 + 4.20i)17-s + (6.21 − 3.97i)18-s + (0.986 + 1.70i)19-s + ⋯
L(s)  = 1  + (−0.869 + 1.50i)2-s + (−0.236 + 0.971i)3-s + (−1.01 − 1.75i)4-s − 1.63·5-s + (−1.25 − 1.20i)6-s + 1.78·8-s + (−0.887 − 0.460i)9-s + (1.42 − 2.46i)10-s + 0.122·11-s + (1.94 − 0.569i)12-s + (0.0675 − 0.116i)13-s + (0.387 − 1.58i)15-s + (−0.540 + 0.936i)16-s + (−0.588 + 1.01i)17-s + (1.46 − 0.937i)18-s + (0.226 + 0.392i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.940 - 0.339i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.940 - 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188674 + 0.0330535i\)
\(L(\frac12)\) \(\approx\) \(0.188674 + 0.0330535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.410 - 1.68i)T \)
7 \( 1 \)
good2 \( 1 + (1.23 - 2.13i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.65T + 5T^{2} \)
11 \( 1 - 0.406T + 11T^{2} \)
13 \( 1 + (-0.243 + 0.421i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.42 - 4.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.986 - 1.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.64T + 23T^{2} \)
29 \( 1 + (3.82 + 6.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.51 + 6.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.16 + 2.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.75 - 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.16 - 2.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.15 + 5.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.78 + 3.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.05 + 5.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.01 + 6.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.80 + 3.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + (0.986 - 1.70i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.08 - 7.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.08 + 10.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.41 + 12.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.74 - 8.21i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.03025271906017686736732402430, −10.00809635676352972290209632035, −9.090435938485731568441401353305, −8.316423318925095352563301663318, −7.69934360584636792567278201125, −6.64818016478749270598421169827, −5.64254075645164944242617004020, −4.52966008268521748185290301699, −3.63558629890152303414607896784, −0.20419701791427244713212913465, 1.10231841662325580025395473489, 2.70233070773490066070666169795, 3.65319047261123497475738269340, 4.99087237038521566401730870619, 7.03442602575293904488512094338, 7.48490753542169378385771416556, 8.673156212162793056711325701805, 9.016457034649848750684573235779, 10.64582849425956196619122537410, 11.18826010795550477915242453897

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