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Properties

Label 2-21e2-9.4-c1-0-18
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

\(L(1)\) \(\approx\) \(1.45082 + 0.121287i\)
\(L(\frac12)\) \(\approx\) \(1.45082 + 0.121287i\)
\(L(\frac{3}{2})\) 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

−11.07029306681072478075835030587, −10.45532279464124650496813008043, −9.539438895332028165422713006229, −8.179801246875899826080383286991, −7.13090511822536748905605504196, −6.30408324044949598339033360202, −5.39983609869076036588736022474, −4.88537802414448571713307736266, −3.30900031645613801471055842742, −1.08637519871768392925950430588, 1.50948830560387669301440078546, 2.88621225477121044351092735901, 4.36527735313029493139939217340, 5.07333652283792946405091584887, 6.48838355619880009931429991899, 7.24569052719994498786223310542, 8.162964195501716387699103589792, 9.725937093845909830484652582257, 10.44145692439900791221572583273, 11.32506577611526247797564276217

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