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Properties

Label 2-21e2-9.4-c1-0-13
Degree $2$
Conductor $441$
Sign $0.0816 - 0.996i$
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  + (0.247 + 0.429i)2-s + (1.37 + 1.05i)3-s + (0.877 − 1.51i)4-s + (−1.84 + 3.19i)5-s + (−0.109 + 0.851i)6-s + 1.86·8-s + (0.792 + 2.89i)9-s − 1.83·10-s + (0.446 + 0.772i)11-s + (2.80 − 1.17i)12-s + (−0.598 + 1.03i)13-s + (−5.90 + 2.46i)15-s + (−1.29 − 2.23i)16-s − 0.249·17-s + (−1.04 + 1.05i)18-s + 2.80·19-s + ⋯
L(s)  = 1  + (0.175 + 0.303i)2-s + (0.795 + 0.606i)3-s + (0.438 − 0.759i)4-s + (−0.825 + 1.43i)5-s + (−0.0447 + 0.347i)6-s + 0.658·8-s + (0.264 + 0.964i)9-s − 0.579·10-s + (0.134 + 0.233i)11-s + (0.809 − 0.337i)12-s + (−0.165 + 0.287i)13-s + (−1.52 + 0.636i)15-s + (−0.323 − 0.559i)16-s − 0.0606·17-s + (−0.246 + 0.249i)18-s + 0.644·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0816 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0816 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0816 - 0.996i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.0816 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41567 + 1.30449i\)
\(L(\frac12)\) \(\approx\) \(1.41567 + 1.30449i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.37 - 1.05i)T \)
7 \( 1 \)
good2 \( 1 + (-0.247 - 0.429i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.84 - 3.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.446 - 0.772i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.598 - 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.249T + 17T^{2} \)
19 \( 1 - 2.80T + 19T^{2} \)
23 \( 1 + (1.23 - 2.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.07 - 3.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.73T + 37T^{2} \)
41 \( 1 + (-2.39 + 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.98 + 8.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.08 + 8.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.88T + 53T^{2} \)
59 \( 1 + (-0.906 + 1.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.40 - 9.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.514 - 0.891i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + 1.83T + 73T^{2} \)
79 \( 1 + (-0.899 - 1.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.16 + 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.40T + 89T^{2} \)
97 \( 1 + (5.52 + 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.21389010432718974992800755220, −10.30669034536320633699478064825, −9.851140935701137624373067391819, −8.538792256760748964709744771632, −7.33619995945276213862387415410, −7.02328225195842211491847694947, −5.66841003433102553596707366860, −4.39228535903513058703384262193, −3.34579123972321004116346480109, −2.21528388608458577474396997482, 1.17278024141915365176269407644, 2.74861061824448536008079892529, 3.84172644487950478505789308201, 4.75314233607271916099760193696, 6.40292874316780386603588993402, 7.63108609875043951574443726508, 8.092551829425010414274588580939, 8.791934345113698643116943724400, 9.824769735535046067043720838028, 11.39049366331349389796414199557

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