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Properties

Label 2-21e2-63.13-c2-0-22
Degree $2$
Conductor $441$
Sign $-0.812 + 0.582i$
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  + (1.91 + 3.32i)2-s + (2.23 + 1.99i)3-s + (−5.35 + 9.27i)4-s + (0.187 + 0.108i)5-s + (−2.34 + 11.2i)6-s − 25.7·8-s + (1.02 + 8.94i)9-s + 0.830i·10-s + (−2.22 − 3.85i)11-s + (−30.5 + 10.0i)12-s + (14.9 + 8.61i)13-s + (0.203 + 0.616i)15-s + (−27.9 − 48.4i)16-s + 5.38i·17-s + (−27.7 + 20.5i)18-s − 7.94i·19-s + ⋯
L(s)  = 1  + (0.959 + 1.66i)2-s + (0.746 + 0.665i)3-s + (−1.33 + 2.31i)4-s + (0.0375 + 0.0216i)5-s + (−0.390 + 1.87i)6-s − 3.21·8-s + (0.113 + 0.993i)9-s + 0.0830i·10-s + (−0.202 − 0.350i)11-s + (−2.54 + 0.839i)12-s + (1.14 + 0.663i)13-s + (0.0135 + 0.0411i)15-s + (−1.74 − 3.02i)16-s + 0.316i·17-s + (−1.54 + 1.14i)18-s − 0.418i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.812 + 0.582i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ -0.812 + 0.582i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.964974 - 3.00296i\)
\(L(\frac12)\) \(\approx\) \(0.964974 - 3.00296i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.23 - 1.99i)T \)
7 \( 1 \)
good2 \( 1 + (-1.91 - 3.32i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + (-0.187 - 0.108i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.22 + 3.85i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-14.9 - 8.61i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 5.38iT - 289T^{2} \)
19 \( 1 + 7.94iT - 361T^{2} \)
23 \( 1 + (-9.53 + 16.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-3.57 - 6.18i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-20.0 - 11.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 10.3T + 1.36e3T^{2} \)
41 \( 1 + (-3.32 - 1.91i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-30.1 - 52.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-42.4 + 24.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 50.0T + 2.80e3T^{2} \)
59 \( 1 + (-75.2 - 43.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (35.6 - 20.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (32.1 - 55.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 11.8T + 5.04e3T^{2} \)
73 \( 1 - 32.0iT - 5.32e3T^{2} \)
79 \( 1 + (7.26 + 12.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-94.9 + 54.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 128. iT - 7.92e3T^{2} \)
97 \( 1 + (-63.3 + 36.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.63194332139017017290164842671, −10.42275894541452624627334057343, −9.043873393771725745468951015722, −8.576554139105132811190317193732, −7.75669192143904737440201481884, −6.66640930640397380947761420078, −5.82448360690835027465285696614, −4.66968636655980956961334713212, −3.98049463597930158534705454164, −2.88223109475429604315914074166, 0.952870871289612051552688547398, 2.06280816068259840774806166255, 3.19460469489594074485230612922, 3.93360342593390187362007918905, 5.31565571609271954234584009191, 6.26064929821923125650381642569, 7.73891816488263827204872343560, 8.922271936107180335302758559097, 9.648869322508740757538308016618, 10.60084019209356519572629300092

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