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Properties

Label 2-21e2-63.4-c1-0-13
Degree $2$
Conductor $441$
Sign $0.842 - 0.538i$
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 + (−1.59 + 0.667i)3-s + (0.877 + 1.51i)4-s + 3.69·5-s + (−0.109 + 0.851i)6-s + 1.86·8-s + (2.10 − 2.13i)9-s + (0.915 − 1.58i)10-s − 0.892·11-s + (−2.41 − 1.84i)12-s + (−0.598 + 1.03i)13-s + (−5.90 + 2.46i)15-s + (−1.29 + 2.23i)16-s + (0.124 − 0.216i)17-s + (−0.393 − 1.43i)18-s + (−1.40 − 2.43i)19-s + ⋯
L(s)  = 1  + (0.175 − 0.303i)2-s + (−0.922 + 0.385i)3-s + (0.438 + 0.759i)4-s + 1.65·5-s + (−0.0447 + 0.347i)6-s + 0.658·8-s + (0.703 − 0.711i)9-s + (0.289 − 0.501i)10-s − 0.269·11-s + (−0.697 − 0.531i)12-s + (−0.165 + 0.287i)13-s + (−1.52 + 0.636i)15-s + (−0.323 + 0.559i)16-s + (0.0303 − 0.0525i)17-s + (−0.0926 − 0.338i)18-s + (−0.322 − 0.557i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.842 - 0.538i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61048 + 0.470652i\)
\(L(\frac12)\) \(\approx\) \(1.61048 + 0.470652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.59 - 0.667i)T \)
7 \( 1 \)
good2 \( 1 + (-0.247 + 0.429i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3.69T + 5T^{2} \)
11 \( 1 + 0.892T + 11T^{2} \)
13 \( 1 + (0.598 - 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.124 + 0.216i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.40 + 2.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
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 \( 1 + (2.36 + 4.09i)T + (-18.5 + 32.0i)T^{2} \)
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 \( 1 + (4.94 - 8.56i)T + (-26.5 - 45.8i)T^{2} \)
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 \( 1 + (-0.915 + 1.58i)T + (-36.5 - 63.2i)T^{2} \)
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 \( 1 + (-1.20 - 2.08i)T + (-44.5 + 77.0i)T^{2} \)
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

−11.00990307405077073243410591759, −10.54814040409818948571801056643, −9.606926087885443217493660655857, −8.738403560704325938482156645128, −7.15880774286087749015060989169, −6.47286181666161188654074433809, −5.44830216517239354056955716876, −4.53876241901627713714388415983, −3.01926221001700523568651192945, −1.73363410706339068500183610619, 1.32429596099997915887046726363, 2.39395959018400465555481652163, 4.81137805539670338530925593083, 5.52241065748632341435235930376, 6.25548198439747543778653290690, 6.83569446459648529640013252406, 8.128316925212301194710902641876, 9.774129783649272468932200549523, 10.05947060887445059832314377158, 10.92713020763194105460946606918

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