Properties

Label 2-21e2-49.11-c1-0-15
Degree $2$
Conductor $441$
Sign $0.0600 + 0.998i$
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  + (−1.90 + 1.76i)2-s + (0.353 − 4.72i)4-s + (1.29 − 3.29i)5-s + (−2.54 + 0.706i)7-s + (4.42 + 5.54i)8-s + (3.35 + 8.55i)10-s + (−0.000902 − 0.000278i)11-s + (0.557 + 2.44i)13-s + (3.60 − 5.84i)14-s + (−8.84 − 1.33i)16-s + (−1.28 + 0.875i)17-s + (−4.05 − 7.02i)19-s + (−15.1 − 7.27i)20-s + (0.00220 − 0.00106i)22-s + (−0.736 − 0.502i)23-s + ⋯
L(s)  = 1  + (−1.34 + 1.24i)2-s + (0.176 − 2.36i)4-s + (0.578 − 1.47i)5-s + (−0.963 + 0.267i)7-s + (1.56 + 1.96i)8-s + (1.06 + 2.70i)10-s + (−0.000272 − 8.39e−5i)11-s + (0.154 + 0.676i)13-s + (0.962 − 1.56i)14-s + (−2.21 − 0.333i)16-s + (−0.311 + 0.212i)17-s + (−0.930 − 1.61i)19-s + (−3.37 − 1.62i)20-s + (0.000471 − 0.000226i)22-s + (−0.153 − 0.104i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0600 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0600 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0600 + 0.998i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.0600 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.247754 - 0.233288i\)
\(L(\frac12)\) \(\approx\) \(0.247754 - 0.233288i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 - 0.706i)T \)
good2 \( 1 + (1.90 - 1.76i)T + (0.149 - 1.99i)T^{2} \)
5 \( 1 + (-1.29 + 3.29i)T + (-3.66 - 3.40i)T^{2} \)
11 \( 1 + (0.000902 + 0.000278i)T + (9.08 + 6.19i)T^{2} \)
13 \( 1 + (-0.557 - 2.44i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (1.28 - 0.875i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (4.05 + 7.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.736 + 0.502i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (6.27 + 3.02i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (-0.104 + 0.180i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.617 + 8.24i)T + (-36.5 + 5.51i)T^{2} \)
41 \( 1 + (1.75 + 2.20i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (6.37 - 7.99i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.166 + 0.154i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (-0.614 + 8.19i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (0.205 + 0.523i)T + (-43.2 + 40.1i)T^{2} \)
61 \( 1 + (-0.734 - 9.80i)T + (-60.3 + 9.09i)T^{2} \)
67 \( 1 + (1.88 - 3.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.36 - 4.02i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (6.17 + 5.73i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (5.86 + 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.21 - 5.32i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-10.3 + 3.18i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−10.40057060242166538932420590452, −9.460348252455171633724378064608, −9.026384370836189600137676079566, −8.502552312444416816901104633267, −7.22224522057193771739146530438, −6.33982719750119833414516877794, −5.59747046219269586909129987628, −4.49532527670174874892763299181, −1.93979895006735516001277804882, −0.31589708437356655380057886719, 1.87102468332412035896172440144, 3.02395070501005302056261463072, 3.67449285300241867913357632797, 6.04242998793141281211864162190, 6.95856576705293074034896493397, 7.86627216463839060686483367466, 8.944285852883731565593853621425, 9.997585826116707999328902466460, 10.26973427765573761824367202483, 10.91513144309006381074950764868

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