Properties

Label 2-21e2-441.268-c1-0-31
Degree $2$
Conductor $441$
Sign $0.991 - 0.129i$
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  + (1.22 + 0.835i)2-s + (0.435 − 1.67i)3-s + (0.0738 + 0.188i)4-s + (−0.839 + 3.67i)5-s + (1.93 − 1.69i)6-s + (2.50 − 0.847i)7-s + (0.593 − 2.60i)8-s + (−2.62 − 1.45i)9-s + (−4.10 + 3.80i)10-s + (3.06 − 1.47i)11-s + (0.347 − 0.0419i)12-s + (3.79 + 2.58i)13-s + (3.78 + 1.05i)14-s + (5.80 + 3.00i)15-s + (3.19 − 2.96i)16-s + (−1.28 + 3.26i)17-s + ⋯
L(s)  = 1  + (0.867 + 0.591i)2-s + (0.251 − 0.967i)3-s + (0.0369 + 0.0941i)4-s + (−0.375 + 1.64i)5-s + (0.790 − 0.690i)6-s + (0.947 − 0.320i)7-s + (0.209 − 0.919i)8-s + (−0.873 − 0.486i)9-s + (−1.29 + 1.20i)10-s + (0.922 − 0.444i)11-s + (0.100 − 0.0120i)12-s + (1.05 + 0.717i)13-s + (1.01 + 0.282i)14-s + (1.49 + 0.776i)15-s + (0.799 − 0.742i)16-s + (−0.310 + 0.791i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.991 - 0.129i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.991 - 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39474 + 0.155835i\)
\(L(\frac12)\) \(\approx\) \(2.39474 + 0.155835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.435 + 1.67i)T \)
7 \( 1 + (-2.50 + 0.847i)T \)
good2 \( 1 + (-1.22 - 0.835i)T + (0.730 + 1.86i)T^{2} \)
5 \( 1 + (0.839 - 3.67i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-3.06 + 1.47i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-3.79 - 2.58i)T + (4.74 + 12.1i)T^{2} \)
17 \( 1 + (1.28 - 3.26i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (-2.34 + 4.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.10 - 3.89i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (2.21 - 0.334i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 + (-2.86 + 4.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.60 - 1.29i)T + (35.3 - 10.9i)T^{2} \)
41 \( 1 + (3.21 - 0.991i)T + (33.8 - 23.0i)T^{2} \)
43 \( 1 + (2.48 + 0.766i)T + (35.5 + 24.2i)T^{2} \)
47 \( 1 + (-5.44 - 3.70i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (4.51 + 0.680i)T + (50.6 + 15.6i)T^{2} \)
59 \( 1 + (9.08 + 2.80i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (1.82 - 4.65i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (7.14 - 12.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.88 + 7.37i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (0.305 + 4.07i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + (0.616 + 1.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.61 - 3.14i)T + (30.3 - 77.2i)T^{2} \)
89 \( 1 + (1.11 - 0.761i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 + (6.47 - 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.37594876013224038819175764284, −10.57290114885147665402869275086, −9.147809110386615579267881875602, −7.987234122343065291474162212144, −7.11253507186003871788263184561, −6.53700954218460569976062277492, −5.81094998608975659525091472432, −4.13260710883046464146370816895, −3.33837011245404330399214003601, −1.59825463949783999981701693094, 1.68297675061043086794280840049, 3.43813557299664027625600121492, 4.30974237300475436332442904681, 4.93757802699816033681010089010, 5.68517528128006089943614507031, 7.915185405955900921772588557604, 8.548494249412090292339794121039, 9.091454818998404820627860934419, 10.39236289997245572731629733836, 11.40519683492644517187224454850

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