Properties

Label 2-21e2-63.38-c1-0-1
Degree $2$
Conductor $441$
Sign $0.999 + 0.0367i$
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  − 2.59i·2-s + (−1.34 − 1.09i)3-s − 4.72·4-s + (−0.626 + 1.08i)5-s + (−2.84 + 3.47i)6-s + 7.07i·8-s + (0.599 + 2.93i)9-s + (2.81 + 1.62i)10-s + (−0.534 + 0.308i)11-s + (6.34 + 5.17i)12-s + (−1.06 + 0.613i)13-s + (2.02 − 0.769i)15-s + 8.88·16-s + (−2.21 + 3.83i)17-s + (7.62 − 1.55i)18-s + (1.64 − 0.950i)19-s + ⋯
L(s)  = 1  − 1.83i·2-s + (−0.774 − 0.632i)3-s − 2.36·4-s + (−0.280 + 0.485i)5-s + (−1.16 + 1.42i)6-s + 2.50i·8-s + (0.199 + 0.979i)9-s + (0.889 + 0.513i)10-s + (−0.161 + 0.0929i)11-s + (1.83 + 1.49i)12-s + (−0.294 + 0.170i)13-s + (0.523 − 0.198i)15-s + 2.22·16-s + (−0.537 + 0.930i)17-s + (1.79 − 0.366i)18-s + (0.377 − 0.218i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0367i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.999 + 0.0367i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.288229 - 0.00530331i\)
\(L(\frac12)\) \(\approx\) \(0.288229 - 0.00530331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.34 + 1.09i)T \)
7 \( 1 \)
good2 \( 1 + 2.59iT - 2T^{2} \)
5 \( 1 + (0.626 - 1.08i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.534 - 0.308i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.06 - 0.613i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.21 - 3.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.64 + 0.950i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.11 + 2.37i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.07 + 2.93i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.48iT - 31T^{2} \)
37 \( 1 + (-1.33 - 2.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.09 - 3.63i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.24 - 3.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.61T + 47T^{2} \)
53 \( 1 + (2.67 + 1.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.56T + 59T^{2} \)
61 \( 1 - 14.4iT - 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (9.95 + 5.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.03T + 79T^{2} \)
83 \( 1 + (-4.36 + 7.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.811 + 1.40i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.76 + 5.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.34513286025109972425122940433, −10.49554584806367700799108202517, −9.821231225313953241721189115337, −8.604273048918712490604983557811, −7.57302565341247696271702815619, −6.30923522021274422826321453350, −5.03179047291212372492823053251, −4.01857812999223005696508685715, −2.69268092309375445616880933042, −1.55035418268486757192659673714, 0.20763361549790341593987244868, 3.80569675637018367650430107868, 4.82066234745612713675703401916, 5.42704133323961923424439475452, 6.38274637281733637875714355623, 7.30633032477359559727269505708, 8.204670711209490087828197831774, 9.223752243357368211145164337674, 9.789120227983131521240027650153, 11.07021082047027102861510486255

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