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Properties

Label 2-21e2-49.22-c1-0-17
Degree $2$
Conductor $441$
Sign $0.132 + 0.991i$
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  + (1.11 − 0.535i)2-s + (−0.298 + 0.373i)4-s + (0.661 − 2.89i)5-s + (−0.553 − 2.58i)7-s + (−0.680 + 2.98i)8-s + (−0.816 − 3.57i)10-s + (4.64 − 2.23i)11-s + (−0.946 + 0.455i)13-s + (−2.00 − 2.57i)14-s + (0.626 + 2.74i)16-s + (−3.51 − 4.40i)17-s − 1.28·19-s + (0.886 + 1.11i)20-s + (3.96 − 4.97i)22-s + (2.10 − 2.64i)23-s + ⋯
L(s)  = 1  + (0.785 − 0.378i)2-s + (−0.149 + 0.186i)4-s + (0.295 − 1.29i)5-s + (−0.209 − 0.977i)7-s + (−0.240 + 1.05i)8-s + (−0.258 − 1.13i)10-s + (1.40 − 0.674i)11-s + (−0.262 + 0.126i)13-s + (−0.534 − 0.689i)14-s + (0.156 + 0.686i)16-s + (−0.852 − 1.06i)17-s − 0.294·19-s + (0.198 + 0.248i)20-s + (0.845 − 1.06i)22-s + (0.439 − 0.551i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46794 - 1.28454i\)
\(L(\frac12)\) \(\approx\) \(1.46794 - 1.28454i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.553 + 2.58i)T \)
good2 \( 1 + (-1.11 + 0.535i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (-0.661 + 2.89i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-4.64 + 2.23i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.946 - 0.455i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (3.51 + 4.40i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + 1.28T + 19T^{2} \)
23 \( 1 + (-2.10 + 2.64i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-1.82 - 2.28i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 4.40T + 31T^{2} \)
37 \( 1 + (-5.04 - 6.32i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-0.169 + 0.743i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-2.19 - 9.60i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (1.72 - 0.832i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-3.32 + 4.16i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-2.34 - 10.2i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-7.10 - 8.90i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 9.22T + 67T^{2} \)
71 \( 1 + (8.91 - 11.1i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-1.30 - 0.626i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + (13.8 + 6.65i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-5.81 - 2.79i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−11.36114956162837311185358554887, −9.975238804616757509425360540983, −8.957213702653159746686346726372, −8.508547312658100117944403790556, −7.09196406194225148589876992730, −5.98864547992608877771737238896, −4.60130861773713141612526827200, −4.36131420634495749719414709114, −2.94435428789405401788884283829, −1.07321319996635127419129491519, 2.13738246307047212325269570346, 3.50642126013559647872599697413, 4.54358795752161761898809639686, 5.90697758767787157606574427337, 6.41403737981326592735512452205, 7.16831068797843774854861013861, 8.775555381154850941725643831058, 9.581797981684032755753796076966, 10.37101703319468671755917771861, 11.42969062292519488996566464302

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