Properties

Label 2-21e2-63.4-c1-0-19
Degree $2$
Conductor $441$
Sign $0.0644 - 0.997i$
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  + (−0.119 + 0.207i)2-s + (0.578 + 1.63i)3-s + (0.971 + 1.68i)4-s + 2.59·5-s + (−0.407 − 0.0753i)6-s − 0.942·8-s + (−2.33 + 1.88i)9-s + (−0.309 + 0.536i)10-s + 4.18·11-s + (−2.18 + 2.55i)12-s + (1.84 − 3.18i)13-s + (1.5 + 4.23i)15-s + (−1.83 + 3.16i)16-s + (0.855 − 1.48i)17-s + (−0.112 − 0.708i)18-s + (−3.57 − 6.19i)19-s + ⋯
L(s)  = 1  + (−0.0845 + 0.146i)2-s + (0.334 + 0.942i)3-s + (0.485 + 0.841i)4-s + 1.15·5-s + (−0.166 − 0.0307i)6-s − 0.333·8-s + (−0.776 + 0.629i)9-s + (−0.0979 + 0.169i)10-s + 1.26·11-s + (−0.630 + 0.738i)12-s + (0.510 − 0.884i)13-s + (0.387 + 1.09i)15-s + (−0.457 + 0.792i)16-s + (0.207 − 0.359i)17-s + (−0.0265 − 0.166i)18-s + (−0.820 − 1.42i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0644 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0644 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0644 - 0.997i$
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: \((2,\ 441,\ (\ :1/2),\ 0.0644 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43263 + 1.34305i\)
\(L(\frac12)\) \(\approx\) \(1.43263 + 1.34305i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.578 - 1.63i)T \)
7 \( 1 \)
good2 \( 1 + (0.119 - 0.207i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.59T + 5T^{2} \)
11 \( 1 - 4.18T + 11T^{2} \)
13 \( 1 + (-1.84 + 3.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.855 + 1.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.57 + 6.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.12T + 23T^{2} \)
29 \( 1 + (-1.06 - 1.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.26 + 5.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.830 + 1.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.10 - 8.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.830 - 1.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.66 + 8.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.32 - 9.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.03 - 5.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.99 - 6.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.13 + 7.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.23T + 71T^{2} \)
73 \( 1 + (-3.57 + 6.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.91 + 8.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.44 - 5.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.51 + 4.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.53 - 2.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.19874393959596110558829808996, −10.39087138477786870167957343295, −9.374519579667329336305932226004, −8.843983880170757442026846307869, −7.83255349995701157096044162838, −6.54505608650711578159711974431, −5.77080371103831951862334506258, −4.39332720695937851774180113418, −3.30539016556965815787328486536, −2.19004695094396587508902487934, 1.55742785345645314644683549015, 1.99049686683857454453441801538, 3.76737561713083108805204072759, 5.64625658469392418979511545998, 6.30035011977435095882614188409, 6.81762333365191124590253469241, 8.293567616212814685624969176766, 9.209295843363214163845627370851, 9.910986619330593101485200074206, 10.88410082701143924004578898032

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