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Properties

Label 2-21e2-9.4-c1-0-25
Degree $2$
Conductor $441$
Sign $0.986 - 0.166i$
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  + (0.551 + 0.955i)2-s + (1.67 − 0.441i)3-s + (0.391 − 0.678i)4-s + (−0.0527 + 0.0913i)5-s + (1.34 + 1.35i)6-s + 3.07·8-s + (2.60 − 1.47i)9-s − 0.116·10-s + (−1.66 − 2.89i)11-s + (0.356 − 1.30i)12-s + (−1.23 + 2.14i)13-s + (−0.0479 + 0.176i)15-s + (0.909 + 1.57i)16-s + 1.61·17-s + (2.85 + 1.67i)18-s − 7.68·19-s + ⋯
L(s)  = 1  + (0.389 + 0.675i)2-s + (0.966 − 0.255i)3-s + (0.195 − 0.339i)4-s + (−0.0235 + 0.0408i)5-s + (0.549 + 0.553i)6-s + 1.08·8-s + (0.869 − 0.493i)9-s − 0.0367·10-s + (−0.503 − 0.871i)11-s + (0.102 − 0.378i)12-s + (−0.343 + 0.595i)13-s + (−0.0123 + 0.0455i)15-s + (0.227 + 0.393i)16-s + 0.391·17-s + (0.672 + 0.395i)18-s − 1.76·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.986 - 0.166i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.986 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47775 + 0.207137i\)
\(L(\frac12)\) \(\approx\) \(2.47775 + 0.207137i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 + 0.441i)T \)
7 \( 1 \)
good2 \( 1 + (-0.551 - 0.955i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.0527 - 0.0913i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.66 + 2.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.23 - 2.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 + 7.68T + 19T^{2} \)
23 \( 1 + (-0.948 + 1.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.64 - 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.63 - 8.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.98T + 37T^{2} \)
41 \( 1 + (-3.74 + 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.77 + 6.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.59 - 2.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.97T + 53T^{2} \)
59 \( 1 + (2.22 - 3.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.83 - 4.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.98 - 8.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 + 4.72T + 73T^{2} \)
79 \( 1 + (3.84 + 6.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.584 + 1.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.02T + 89T^{2} \)
97 \( 1 + (1.90 + 3.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.83617940035582941852535850469, −10.40720700344645144677120742609, −9.009893512563144414397804970606, −8.415919317915407257522515481708, −7.21749469603022890485208843888, −6.70677264320379450663336139963, −5.49425216058743207020287599207, −4.39938698964200581815126924842, −3.06327578362875273827831529922, −1.66011332540871236667406239332, 2.06206203137752319870822918761, 2.84108668805684756602952556524, 4.09667199259636823452661288303, 4.78948273451410471648567028886, 6.55083937491059017527178635924, 7.83098709514082397235453165356, 8.091197727120531063776116324732, 9.520373488919040260599877453109, 10.25729827715696281724283178723, 10.99961165861826115833004499458

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