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Properties

Label 2-21e2-7.2-c1-0-11
Degree $2$
Conductor $441$
Sign $0.827 + 0.561i$
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  + (0.207 − 0.358i)2-s + (0.914 + 1.58i)4-s + (1.70 − 2.95i)5-s + 1.58·8-s + (−0.707 − 1.22i)10-s + (−1 − 1.73i)11-s + 2.58·13-s + (−1.49 + 2.59i)16-s + (−1.12 − 1.94i)17-s + (1.41 − 2.44i)19-s + 6.24·20-s − 0.828·22-s + (−3.82 + 6.63i)23-s + (−3.32 − 5.76i)25-s + (0.535 − 0.927i)26-s + ⋯
L(s)  = 1  + (0.146 − 0.253i)2-s + (0.457 + 0.791i)4-s + (0.763 − 1.32i)5-s + 0.560·8-s + (−0.223 − 0.387i)10-s + (−0.301 − 0.522i)11-s + 0.717·13-s + (−0.374 + 0.649i)16-s + (−0.271 − 0.471i)17-s + (0.324 − 0.561i)19-s + 1.39·20-s − 0.176·22-s + (−0.798 + 1.38i)23-s + (−0.665 − 1.15i)25-s + (0.105 − 0.181i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82114 - 0.559583i\)
\(L(\frac12)\) \(\approx\) \(1.82114 - 0.559583i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.207 + 0.358i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.70 + 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
17 \( 1 + (1.12 + 1.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.82 - 6.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.82T + 29T^{2} \)
31 \( 1 + (-0.585 - 1.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.585 + 1.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.12 - 10.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.82 - 4.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.31T + 71T^{2} \)
73 \( 1 + (6.94 + 12.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.82 - 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.31T + 83T^{2} \)
89 \( 1 + (7.12 - 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.58T + 97T^{2} \)
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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

−11.24667281336090943574292072134, −10.14387753450478414463278886894, −9.080966503675722766535044128878, −8.448874331230686604510590835571, −7.48127741693162023665546619960, −6.21021805321040083135888490447, −5.23584235566298110067491687426, −4.14317648491255603552024997444, −2.82041242102347821473887324264, −1.39935063939358553611961969645, 1.80929089991354634401080511054, 2.89347429211605981768832977638, 4.52437959695482811351491481854, 5.90939540950007645522784922413, 6.36778543852952074396382317637, 7.19930000294305810455507696298, 8.390079733707483887601553620923, 9.849362677779063681978777218866, 10.31181188505245573130361625221, 10.90487635463469946891594975151

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