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Properties

Label 2-21e2-147.89-c1-0-12
Degree $2$
Conductor $441$
Sign $0.826 + 0.563i$
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  + (−1.73 + 1.87i)2-s + (−0.339 − 4.52i)4-s + (0.526 + 1.34i)5-s + (2.25 − 1.39i)7-s + (5.07 + 4.04i)8-s + (−3.42 − 1.34i)10-s + (−1.53 − 4.97i)11-s + (−4.91 − 1.12i)13-s + (−1.30 + 6.63i)14-s + (−7.42 + 1.11i)16-s + (−5.11 − 3.48i)17-s + (−3.65 − 2.10i)19-s + (5.88 − 2.83i)20-s + (11.9 + 5.77i)22-s + (1.02 + 1.50i)23-s + ⋯
L(s)  = 1  + (−1.22 + 1.32i)2-s + (−0.169 − 2.26i)4-s + (0.235 + 0.599i)5-s + (0.850 − 0.525i)7-s + (1.79 + 1.42i)8-s + (−1.08 − 0.425i)10-s + (−0.462 − 1.49i)11-s + (−1.36 − 0.311i)13-s + (−0.349 + 1.77i)14-s + (−1.85 + 0.279i)16-s + (−1.23 − 0.845i)17-s + (−0.837 − 0.483i)19-s + (1.31 − 0.634i)20-s + (2.55 + 1.23i)22-s + (0.213 + 0.313i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.826 + 0.563i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.826 + 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.456471 - 0.140825i\)
\(L(\frac12)\) \(\approx\) \(0.456471 - 0.140825i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.25 + 1.39i)T \)
good2 \( 1 + (1.73 - 1.87i)T + (-0.149 - 1.99i)T^{2} \)
5 \( 1 + (-0.526 - 1.34i)T + (-3.66 + 3.40i)T^{2} \)
11 \( 1 + (1.53 + 4.97i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (4.91 + 1.12i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (5.11 + 3.48i)T + (6.21 + 15.8i)T^{2} \)
19 \( 1 + (3.65 + 2.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.02 - 1.50i)T + (-8.40 + 21.4i)T^{2} \)
29 \( 1 + (2.64 + 5.48i)T + (-18.0 + 22.6i)T^{2} \)
31 \( 1 + (2.33 - 1.35i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.262 + 3.50i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (0.957 - 1.20i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-6.06 - 7.60i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-2.72 - 2.52i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (2.32 - 0.174i)T + (52.4 - 7.89i)T^{2} \)
59 \( 1 + (-3.19 + 8.14i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (0.839 + 0.0629i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-1.52 - 2.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.48 - 7.24i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-10.5 - 11.3i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (-1.11 + 1.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.19 + 9.61i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-1.58 - 0.489i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−10.89086503179802032880178813975, −9.938255542376916569426419311457, −9.015084742703978310816666280442, −8.183250578987742563488977729719, −7.41794078258995012590837299373, −6.67573957094193980996260403104, −5.65637452257907893973341011626, −4.66312064149698103725670880916, −2.47668988705440431049484496890, −0.42961336402198876512712225716, 1.81585648040913183030132736264, 2.32104749248211832673431987945, 4.22532965676760656207628894805, 5.09771164233245155943451663972, 7.03698918860969747420885332909, 7.929654654760938269863169423227, 8.872117168330844983082040586615, 9.343941707522415275797185382711, 10.40508574033418758203188071713, 10.92536031929883517808332595228

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