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Properties

Label 2-21e2-147.41-c1-0-0
Degree $2$
Conductor $441$
Sign $-0.468 + 0.883i$
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  + (−0.947 + 0.755i)2-s + (−0.118 + 0.518i)4-s + (−3.15 + 1.52i)5-s + (1.53 + 2.15i)7-s + (−1.33 − 2.76i)8-s + (1.84 − 3.82i)10-s + (0.959 − 0.765i)11-s + (−4.81 + 3.84i)13-s + (−3.08 − 0.883i)14-s + (2.39 + 1.15i)16-s + (0.101 + 0.445i)17-s − 8.04i·19-s + (−0.414 − 1.81i)20-s + (−0.330 + 1.44i)22-s + (1.94 + 0.443i)23-s + ⋯
L(s)  = 1  + (−0.669 + 0.534i)2-s + (−0.0591 + 0.259i)4-s + (−1.41 + 0.679i)5-s + (0.579 + 0.814i)7-s + (−0.470 − 0.977i)8-s + (0.582 − 1.20i)10-s + (0.289 − 0.230i)11-s + (−1.33 + 1.06i)13-s + (−0.823 − 0.236i)14-s + (0.597 + 0.287i)16-s + (0.0246 + 0.107i)17-s − 1.84i·19-s + (−0.0926 − 0.405i)20-s + (−0.0705 + 0.309i)22-s + (0.405 + 0.0924i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.468 + 0.883i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.468 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0846208 - 0.140689i\)
\(L(\frac12)\) \(\approx\) \(0.0846208 - 0.140689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.53 - 2.15i)T \)
good2 \( 1 + (0.947 - 0.755i)T + (0.445 - 1.94i)T^{2} \)
5 \( 1 + (3.15 - 1.52i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.959 + 0.765i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (4.81 - 3.84i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.101 - 0.445i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + 8.04iT - 19T^{2} \)
23 \( 1 + (-1.94 - 0.443i)T + (20.7 + 9.97i)T^{2} \)
29 \( 1 + (6.85 - 1.56i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + 6.87iT - 31T^{2} \)
37 \( 1 + (-0.226 - 0.994i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (5.35 - 2.57i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (8.06 + 3.88i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-0.922 - 1.15i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-7.14 - 1.63i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 + (6.83 + 3.29i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (6.28 - 1.43i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 - 1.41T + 67T^{2} \)
71 \( 1 + (-14.0 - 3.19i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (1.23 + 0.982i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + 6.59T + 79T^{2} \)
83 \( 1 + (3.24 - 4.06i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (8.27 - 10.3i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 - 11.9iT - 97T^{2} \)
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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

−11.65704448459829216987540611764, −11.07598067199472268170589259347, −9.538162590585378072591584767968, −8.912984994023605768811159934205, −7.989518841127700171504403181857, −7.24804283045259386383259880112, −6.65849057122802011877564930721, −4.94072331441900334772174371421, −3.87530079434174689995212799040, −2.62291699003193909343731603711, 0.13206160536326270197333108621, 1.53345495640136626439222735499, 3.43521564040165144497056032360, 4.61089230615424317066153935077, 5.44536953617410189754945615005, 7.25587392902486316926958311804, 7.947725410090430732197123197143, 8.602991171820900638555411871449, 9.831075346401353640781916752299, 10.43154254315177012581580059722

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