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Properties

Label 2-21e2-63.23-c2-0-38
Degree $2$
Conductor $441$
Sign $0.540 + 0.841i$
Analytic cond. $12.0163$
Root an. cond. $3.46646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·2-s + (−1.5 + 2.59i)3-s + 1.00·4-s + (−3 + 1.73i)5-s + (4.5 + 2.59i)6-s − 8.66i·8-s + (−4.5 − 7.79i)9-s + (2.99 + 5.19i)10-s + (1.5 + 0.866i)11-s + (−1.50 + 2.59i)12-s + (2 − 3.46i)13-s − 10.3i·15-s − 10.9·16-s + (13.5 − 7.79i)17-s + (−13.5 + 7.79i)18-s + (−5.5 + 9.52i)19-s + ⋯
L(s)  = 1  − 0.866i·2-s + (−0.5 + 0.866i)3-s + 0.250·4-s + (−0.600 + 0.346i)5-s + (0.750 + 0.433i)6-s − 1.08i·8-s + (−0.5 − 0.866i)9-s + (0.299 + 0.519i)10-s + (0.136 + 0.0787i)11-s + (−0.125 + 0.216i)12-s + (0.153 − 0.266i)13-s − 0.692i·15-s − 0.687·16-s + (0.794 − 0.458i)17-s + (−0.750 + 0.433i)18-s + (−0.289 + 0.501i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.540 + 0.841i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.540 + 0.841i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.28151 - 0.699609i\)
\(L(\frac12)\) \(\approx\) \(1.28151 - 0.699609i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 \)
good2 \( 1 + 1.73iT - 4T^{2} \)
5 \( 1 + (3 - 1.73i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-13.5 + 7.79i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (5.5 - 9.52i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-24 + 13.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-39 + 22.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 32T + 961T^{2} \)
37 \( 1 + (-17 + 29.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (10.5 + 6.06i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-30.5 - 52.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 48.4iT - 2.20e3T^{2} \)
53 \( 1 + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 50.2iT - 3.48e3T^{2} \)
61 \( 1 - 56T + 3.72e3T^{2} \)
67 \( 1 + 31T + 4.48e3T^{2} \)
71 \( 1 + 31.1iT - 5.04e3T^{2} \)
73 \( 1 + (32.5 + 56.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 38T + 6.24e3T^{2} \)
83 \( 1 + (42 - 24.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (108 + 62.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-57.5 - 99.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.79099681125557011835123606332, −10.16994052149051360475148903293, −9.383088613660855499231476493111, −8.115565213032866939296626031528, −6.93678753518574773483949803314, −5.99835311765873498431896863198, −4.65206702712917515910319057020, −3.64508853238292063849600193329, −2.77498822113853578336385968400, −0.791861851556797233383498870812, 1.16713103074793265534927428255, 2.78506120359528270258312601508, 4.60091723265448127653998836105, 5.61831408657135468014299477922, 6.52590256006308433673962941432, 7.22953465595280053776904021674, 8.116718544770657019058916020839, 8.714025859843364598546306762638, 10.34137919385693955204457431290, 11.30704143673092850690197596530

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