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Properties

Label 2-21e2-21.17-c1-0-9
Degree $2$
Conductor $441$
Sign $0.708 + 0.706i$
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  + (2.09 − 1.20i)2-s + (1.91 − 3.31i)4-s + (1.68 + 2.92i)5-s − 4.41i·8-s + (7.06 + 4.07i)10-s + (−0.717 − 0.414i)11-s − 3.37i·13-s + (−1.49 − 2.59i)16-s + (0.699 − 1.21i)17-s + (−5.85 + 3.37i)19-s + 12.9·20-s − 2·22-s + (1.73 − i)23-s + (−3.20 + 5.55i)25-s + (−4.07 − 7.06i)26-s + ⋯
L(s)  = 1  + (1.47 − 0.853i)2-s + (0.957 − 1.65i)4-s + (0.755 + 1.30i)5-s − 1.56i·8-s + (2.23 + 1.28i)10-s + (−0.216 − 0.124i)11-s − 0.937i·13-s + (−0.374 − 0.649i)16-s + (0.169 − 0.293i)17-s + (−1.34 + 0.775i)19-s + 2.89·20-s − 0.426·22-s + (0.361 − 0.208i)23-s + (−0.641 + 1.11i)25-s + (−0.799 − 1.38i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.708 + 0.706i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.708 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.02572 - 1.25069i\)
\(L(\frac12)\) \(\approx\) \(3.02572 - 1.25069i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (-2.09 + 1.20i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.68 - 2.92i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.717 + 0.414i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.37iT - 13T^{2} \)
17 \( 1 + (-0.699 + 1.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.85 - 3.37i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.82iT - 29T^{2} \)
31 \( 1 + (5.85 + 3.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.29 - 2.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.15T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + (-3.37 - 5.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.12 - 3.53i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.37 + 5.85i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.06 + 4.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.24 - 7.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.82iT - 71T^{2} \)
73 \( 1 + (1.21 + 0.699i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.82 - 8.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + (-3.08 - 5.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.39iT - 97T^{2} \)
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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

−11.03607529057306473437092345155, −10.43306781484183561782418828130, −9.853761460057150318623188394841, −8.180223143722964074043853772906, −6.79926002531073783698006434096, −6.03011162132432286115428248309, −5.23459578125511602074475127557, −3.85943493784754625804396508138, −2.91916926838326522823996677275, −2.04170129539380142522836152809, 1.97163582814220057074249002410, 3.69230141838166454613752409805, 4.81540758714355550253417224777, 5.25640621614387521758732946640, 6.36726158530000273498997123025, 7.12398571235663584383247612505, 8.490085271383370662580017402618, 9.102061180327774545434980222182, 10.40466746919374084796427414744, 11.72596260160780490064979604111

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