Properties

Degree $2$
Conductor $441$
Sign $-0.198 - 0.980i$
Motivic weight $1$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.198 - 0.980i$
Motivic weight: \(1\)
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797675 + 0.974976i\)
\(L(\frac12)\) \(\approx\) \(0.797675 + 0.974976i\)
\(L(\frac{3}{2})\) 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

−11.36697967919902650666225095374, −10.73001648597809997030422507855, −9.864663205143531420010452469800, −8.297441275485645495557976386589, −7.66915207771509112154498116576, −6.94689985994893705683983160196, −5.86041074892827605199304610430, −4.13591865704040257148115596797, −3.34012287377486296780811411006, −2.36156477598132384456261576151, 0.74835705711905262724000224294, 2.43344063322604878847884816356, 4.49463792035997144237331156952, 4.84796881351525067302331777052, 6.08411675096087786027221507372, 7.21090942184303427356446428880, 8.042084250611511671050344033018, 9.046873608827497182945386989692, 9.976447265824362015815314759799, 10.83518081566787411249410500048

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