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Properties

Label 2-21e2-63.13-c2-0-42
Degree $2$
Conductor $441$
Sign $0.302 - 0.953i$
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  + (1.41 + 2.45i)2-s + (2.70 − 1.30i)3-s + (−2.01 + 3.49i)4-s + (2.07 + 1.19i)5-s + (7.02 + 4.78i)6-s − 0.0884·8-s + (5.60 − 7.04i)9-s + 6.79i·10-s + (−5.69 − 9.86i)11-s + (−0.896 + 12.0i)12-s + (13.4 + 7.77i)13-s + (7.17 + 0.533i)15-s + (7.93 + 13.7i)16-s + 12.1i·17-s + (25.2 + 3.77i)18-s + 16.0i·19-s + ⋯
L(s)  = 1  + (0.708 + 1.22i)2-s + (0.900 − 0.434i)3-s + (−0.503 + 0.872i)4-s + (0.415 + 0.239i)5-s + (1.17 + 0.797i)6-s − 0.0110·8-s + (0.622 − 0.782i)9-s + 0.679i·10-s + (−0.517 − 0.896i)11-s + (−0.0747 + 1.00i)12-s + (1.03 + 0.597i)13-s + (0.478 + 0.0355i)15-s + (0.496 + 0.859i)16-s + 0.711i·17-s + (1.40 + 0.209i)18-s + 0.842i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.302 - 0.953i$
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: \((2,\ 441,\ (\ :1),\ 0.302 - 0.953i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.07281 + 2.24845i\)
\(L(\frac12)\) \(\approx\) \(3.07281 + 2.24845i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.70 + 1.30i)T \)
7 \( 1 \)
good2 \( 1 + (-1.41 - 2.45i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + (-2.07 - 1.19i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (5.69 + 9.86i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-13.4 - 7.77i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 12.1iT - 289T^{2} \)
19 \( 1 - 16.0iT - 361T^{2} \)
23 \( 1 + (-8.02 + 13.9i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-16.2 - 28.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (36.7 + 21.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 7.22T + 1.36e3T^{2} \)
41 \( 1 + (-7.55 - 4.36i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (22.2 + 38.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (22.8 - 13.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 68.1T + 2.80e3T^{2} \)
59 \( 1 + (-82.4 - 47.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (42.8 - 24.7i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-40.9 + 70.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 112.T + 5.04e3T^{2} \)
73 \( 1 - 67.9iT - 5.32e3T^{2} \)
79 \( 1 + (68.4 + 118. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (18.8 - 10.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 108. iT - 7.92e3T^{2} \)
97 \( 1 + (96.5 - 55.7i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.02099655328373562044998858716, −10.12114978969033855082332559693, −8.741222486832519558526335186377, −8.268074982843073120439423037058, −7.26756120050145131593110217968, −6.32353155094211230873963128525, −5.78015041109981685114066441137, −4.27263639202951085382490172914, −3.29453002358147517299257623805, −1.68044317597475058201896271959, 1.51347423832403732507719484279, 2.63384500531772754545526602012, 3.51973381346497445243283742927, 4.63361652640927256018952979912, 5.39359861589545021799754131725, 7.16676003003977867780031903544, 8.141353057410999501963171570164, 9.344859633412766607775606692072, 9.884844084198086543350057732857, 10.81363338838315319374693259147

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