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Properties

Label 2-21e2-49.15-c1-0-5
Degree $2$
Conductor $441$
Sign $0.991 - 0.129i$
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  + (0.441 − 1.93i)2-s + (−1.75 − 0.843i)4-s + (−2.19 + 2.75i)5-s + (1.51 + 2.16i)7-s + (0.0692 − 0.0868i)8-s + (4.36 + 5.47i)10-s + (−0.927 + 4.06i)11-s + (−1.27 + 5.57i)13-s + (4.86 − 1.97i)14-s + (−2.56 − 3.21i)16-s + (2.79 − 1.34i)17-s − 2.60·19-s + (6.17 − 2.97i)20-s + (7.45 + 3.59i)22-s + (−0.106 − 0.0513i)23-s + ⋯
L(s)  = 1  + (0.312 − 1.36i)2-s + (−0.875 − 0.421i)4-s + (−0.983 + 1.23i)5-s + (0.572 + 0.819i)7-s + (0.0244 − 0.0307i)8-s + (1.38 + 1.73i)10-s + (−0.279 + 1.22i)11-s + (−0.352 + 1.54i)13-s + (1.30 − 0.528i)14-s + (−0.640 − 0.803i)16-s + (0.678 − 0.326i)17-s − 0.596·19-s + (1.38 − 0.665i)20-s + (1.59 + 0.765i)22-s + (−0.0222 − 0.0107i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.991 - 0.129i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.991 - 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30786 + 0.0853712i\)
\(L(\frac12)\) \(\approx\) \(1.30786 + 0.0853712i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.51 - 2.16i)T \)
good2 \( 1 + (-0.441 + 1.93i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (2.19 - 2.75i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (0.927 - 4.06i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.27 - 5.57i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-2.79 + 1.34i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + 2.60T + 19T^{2} \)
23 \( 1 + (0.106 + 0.0513i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-4.91 + 2.36i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 - 8.12T + 31T^{2} \)
37 \( 1 + (2.18 - 1.05i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (2.52 - 3.17i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (6.43 + 8.07i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-0.896 + 3.92i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-4.38 - 2.11i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (-5.79 - 7.26i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (5.49 - 2.64i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 - 2.57T + 67T^{2} \)
71 \( 1 + (-1.85 - 0.894i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.14 - 9.39i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 - 8.80T + 79T^{2} \)
83 \( 1 + (0.504 + 2.21i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (2.16 + 9.48i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−11.43246162272385048079029665677, −10.38953953077037305525886665746, −9.827973338296820790153906674648, −8.543938471871379311803794633097, −7.36454357641857312736497085778, −6.66605168410220086609328030624, −4.83091019214137981084626233525, −4.08760527209439520174036439212, −2.82192849239052759381804083160, −2.01370248255302216301521440344, 0.77744567823049731907052301113, 3.51139196540550329360672563551, 4.71237217954380004897189096316, 5.25879275425356329087026271434, 6.39117790831946512857203031495, 7.71435783116635654942248523582, 8.122874449295242257529597514675, 8.579714210903047081751937462845, 10.26420681493294218658652154642, 11.10369727301524777603566403833

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