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Properties

Label 2-21e2-9.4-c1-0-33
Degree $2$
Conductor $441$
Sign $-0.482 - 0.875i$
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  + (−1.02 − 1.77i)2-s + (0.608 − 1.62i)3-s + (−1.10 + 1.92i)4-s + (0.0731 − 0.126i)5-s + (−3.50 + 0.582i)6-s + 0.446·8-s + (−2.25 − 1.97i)9-s − 0.300·10-s + (−0.832 − 1.44i)11-s + (2.43 + 2.96i)12-s + (0.0999 − 0.173i)13-s + (−0.160 − 0.195i)15-s + (1.75 + 3.04i)16-s − 6.27·17-s + (−1.19 + 6.04i)18-s − 6.91·19-s + ⋯
L(s)  = 1  + (−0.726 − 1.25i)2-s + (0.351 − 0.936i)3-s + (−0.554 + 0.960i)4-s + (0.0327 − 0.0566i)5-s + (−1.43 + 0.237i)6-s + 0.157·8-s + (−0.752 − 0.658i)9-s − 0.0949·10-s + (−0.250 − 0.434i)11-s + (0.704 + 0.856i)12-s + (0.0277 − 0.0480i)13-s + (−0.0415 − 0.0505i)15-s + (0.439 + 0.761i)16-s − 1.52·17-s + (−0.280 + 1.42i)18-s − 1.58·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.482 - 0.875i$
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: \((2,\ 441,\ (\ :1/2),\ -0.482 - 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.312194 + 0.528396i\)
\(L(\frac12)\) \(\approx\) \(0.312194 + 0.528396i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.608 + 1.62i)T \)
7 \( 1 \)
good2 \( 1 + (1.02 + 1.77i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.0731 + 0.126i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.832 + 1.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0999 + 0.173i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.27T + 17T^{2} \)
19 \( 1 + 6.91T + 19T^{2} \)
23 \( 1 + (-3.09 + 5.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.46 + 4.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 + 2.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.00T + 37T^{2} \)
41 \( 1 + (-1.15 + 2.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.940 + 1.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.905 - 1.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.34T + 53T^{2} \)
59 \( 1 + (-2.28 + 3.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.339 - 0.587i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.09 + 5.35i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 - 1.55T + 73T^{2} \)
79 \( 1 + (6.39 + 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.75 - 6.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.06T + 89T^{2} \)
97 \( 1 + (3.98 + 6.90i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.88686197259707170065942034067, −9.548939943901269823108105699515, −8.728943618649517941823529065846, −8.246126673144477799784428487276, −6.88787224309749563938365656449, −6.00012549135124309301471731558, −4.20275584469408144476799647402, −2.79313164998244752711340090053, −2.01971294696847498053798375376, −0.44226973834403217998188942617, 2.57376833261078443603850558268, 4.18186758841195527670819090484, 5.16787718244671099490638283283, 6.31153120140239399412745366478, 7.16782362650960979896070064484, 8.295947695479361768564699185240, 8.838214015640061788913572598118, 9.611263126928359119257964823630, 10.54092518547686596592096340859, 11.33517799667495845156727813803

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