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Properties

Label 2-21e2-63.58-c1-0-30
Degree $2$
Conductor $441$
Sign $0.995 - 0.0932i$
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  + 2.46·2-s + (1.25 + 1.19i)3-s + 4.05·4-s + (−1.82 − 3.16i)5-s + (3.08 + 2.94i)6-s + 5.05·8-s + (0.136 + 2.99i)9-s + (−4.50 − 7.79i)10-s + (−0.203 + 0.351i)11-s + (5.07 + 4.85i)12-s + (−0.243 + 0.421i)13-s + (1.5 − 6.15i)15-s + 4.32·16-s + (2.42 + 4.20i)17-s + (0.336 + 7.37i)18-s + (−0.986 + 1.70i)19-s + ⋯
L(s)  = 1  + 1.73·2-s + (0.723 + 0.690i)3-s + 2.02·4-s + (−0.817 − 1.41i)5-s + (1.25 + 1.20i)6-s + 1.78·8-s + (0.0455 + 0.998i)9-s + (−1.42 − 2.46i)10-s + (−0.0612 + 0.106i)11-s + (1.46 + 1.40i)12-s + (−0.0675 + 0.116i)13-s + (0.387 − 1.58i)15-s + 1.08·16-s + (0.588 + 1.01i)17-s + (0.0792 + 1.73i)18-s + (−0.226 + 0.392i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.995 - 0.0932i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.995 - 0.0932i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.89499 + 0.181942i\)
\(L(\frac12)\) \(\approx\) \(3.89499 + 0.181942i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.25 - 1.19i)T \)
7 \( 1 \)
good2 \( 1 - 2.46T + 2T^{2} \)
5 \( 1 + (1.82 + 3.16i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.203 - 0.351i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.243 - 0.421i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.42 - 4.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.986 - 1.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.32 + 4.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.82 + 6.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.02T + 31T^{2} \)
37 \( 1 + (1.16 - 2.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.75 + 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.16 - 2.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.31T + 47T^{2} \)
53 \( 1 + (-1.78 - 3.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.11T + 59T^{2} \)
61 \( 1 - 8.02T + 61T^{2} \)
67 \( 1 - 3.60T + 67T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + (-0.986 - 1.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 8.16T + 79T^{2} \)
83 \( 1 + (-6.08 - 10.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.41 + 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.74 + 8.21i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.44138003305367584262359098008, −10.48616760664341660816106663514, −9.244030512587919100996570373837, −8.278149195459204939243479758335, −7.51197044247367999228469876392, −5.90099192593819822832984027222, −5.05691643204883505936104235578, −4.10285483253058698338901311643, −3.75344617365386696490603816606, −2.11655428445145476905404738566, 2.31515234506503335878517529124, 3.27128679546607330653989653804, 3.80906418113187744685207224270, 5.35378656515566712391843222720, 6.48812596584846744914422770669, 7.24101186943472857404465722455, 7.71761930651031420784952325102, 9.305228224619624747401781798548, 10.73916450942596149160576764139, 11.43773328073996076199218399373

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