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Properties

Label 2-21e2-63.34-c2-0-30
Degree $2$
Conductor $441$
Sign $0.278 - 0.960i$
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  + (0.198 − 0.343i)2-s + (−2.98 + 0.338i)3-s + (1.92 + 3.32i)4-s + (−2.57 + 1.48i)5-s + (−0.474 + 1.08i)6-s + 3.10·8-s + (8.77 − 2.01i)9-s + 1.17i·10-s + (9.33 − 16.1i)11-s + (−6.85 − 9.27i)12-s + (9.50 − 5.48i)13-s + (7.17 − 5.30i)15-s + (−7.07 + 12.2i)16-s + 6.64i·17-s + (1.04 − 3.40i)18-s + 22.6i·19-s + ⋯
L(s)  = 1  + (0.0990 − 0.171i)2-s + (−0.993 + 0.112i)3-s + (0.480 + 0.832i)4-s + (−0.514 + 0.297i)5-s + (−0.0790 + 0.181i)6-s + 0.388·8-s + (0.974 − 0.224i)9-s + 0.117i·10-s + (0.848 − 1.47i)11-s + (−0.571 − 0.772i)12-s + (0.731 − 0.422i)13-s + (0.478 − 0.353i)15-s + (−0.441 + 0.765i)16-s + 0.390i·17-s + (0.0580 − 0.189i)18-s + 1.19i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.278 - 0.960i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.278 - 0.960i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.06979 + 0.803870i\)
\(L(\frac12)\) \(\approx\) \(1.06979 + 0.803870i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.98 - 0.338i)T \)
7 \( 1 \)
good2 \( 1 + (-0.198 + 0.343i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + (2.57 - 1.48i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-9.33 + 16.1i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-9.50 + 5.48i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 6.64iT - 289T^{2} \)
19 \( 1 - 22.6iT - 361T^{2} \)
23 \( 1 + (-13.4 - 23.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (10.3 - 17.9i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (16.4 - 9.49i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 7.15T + 1.36e3T^{2} \)
41 \( 1 + (-66.6 + 38.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (14.9 - 25.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (5.23 + 3.02i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 39.8T + 2.80e3T^{2} \)
59 \( 1 + (30.4 - 17.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-27.8 - 16.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (0.778 + 1.34i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 111.T + 5.04e3T^{2} \)
73 \( 1 - 42.8iT - 5.32e3T^{2} \)
79 \( 1 + (22.9 - 39.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-31.3 - 18.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 41.5iT - 7.92e3T^{2} \)
97 \( 1 + (-120. - 69.6i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.08443483681304409048498288287, −10.79522203218768749202921062754, −9.301994543472212974944650404723, −8.201056748614547825392923970360, −7.35982792073564285021040577363, −6.33562412696350195608395078174, −5.57005357945282474964703291434, −3.84083310085032177300806197499, −3.46786506170104483518549789093, −1.31942938193745570497200438437, 0.70438499932454905344934318003, 2.02986921393245531099396321943, 4.31173869807903027899685079454, 4.84828638877230148482320605838, 6.16681712468938732230335065853, 6.78982726608510998078508870348, 7.59286250051057939166975247836, 9.174316118996764281202629281413, 9.890711276785738591054556643885, 10.99705057832461761588705561849

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