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Properties

Label 2-21e2-49.15-c1-0-13
Degree $2$
Conductor $441$
Sign $0.0818 + 0.996i$
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  + (0.345 − 1.51i)2-s + (−0.370 − 0.178i)4-s + (−0.0888 + 0.111i)5-s + (−0.524 + 2.59i)7-s + (1.53 − 1.92i)8-s + (0.137 + 0.172i)10-s + (0.506 − 2.21i)11-s + (1.21 − 5.30i)13-s + (3.74 + 1.68i)14-s + (−2.90 − 3.63i)16-s + (4.14 − 1.99i)17-s + 7.00·19-s + (0.0528 − 0.0254i)20-s + (−3.18 − 1.53i)22-s + (−2.32 − 1.11i)23-s + ⋯
L(s)  = 1  + (0.244 − 1.07i)2-s + (−0.185 − 0.0892i)4-s + (−0.0397 + 0.0498i)5-s + (−0.198 + 0.980i)7-s + (0.543 − 0.681i)8-s + (0.0436 + 0.0546i)10-s + (0.152 − 0.668i)11-s + (0.335 − 1.47i)13-s + (1.00 + 0.451i)14-s + (−0.725 − 0.909i)16-s + (1.00 − 0.484i)17-s + 1.60·19-s + (0.0118 − 0.00568i)20-s + (−0.678 − 0.326i)22-s + (−0.484 − 0.233i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0818 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0818 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0818 + 0.996i$
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: \((2,\ 441,\ (\ :1/2),\ 0.0818 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30464 - 1.20191i\)
\(L(\frac12)\) \(\approx\) \(1.30464 - 1.20191i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.524 - 2.59i)T \)
good2 \( 1 + (-0.345 + 1.51i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (0.0888 - 0.111i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (-0.506 + 2.21i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-1.21 + 5.30i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-4.14 + 1.99i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 - 7.00T + 19T^{2} \)
23 \( 1 + (2.32 + 1.11i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (6.29 - 3.03i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 - 5.89T + 31T^{2} \)
37 \( 1 + (4.97 - 2.39i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (3.23 - 4.06i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-0.0515 - 0.0646i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (2.01 - 8.84i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (6.33 + 3.05i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (-4.27 - 5.35i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (3.89 - 1.87i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 - 1.03T + 67T^{2} \)
71 \( 1 + (-0.359 - 0.172i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (0.338 + 1.48i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + (0.655 + 2.87i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-2.54 - 11.1i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + 3.70T + 97T^{2} \)
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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

−11.13625844981469388185795802426, −10.10724625535108925738128354795, −9.443076349595681293565094105706, −8.237086525371534227160659236453, −7.32398843853539441647234117880, −5.91362724231913056380602642961, −5.12025066714710848895053340890, −3.30107933811868770988778150248, −3.03173379736180662869361649969, −1.25279645247069633590328685092, 1.69654417409486120820063252222, 3.72302771311286947180941993963, 4.69400250581625387027498433849, 5.82588894349383960573047392716, 6.81490590134197572918217533444, 7.38403752079026500088589187626, 8.273498966790462109949941216014, 9.557120840724310059267754363831, 10.29298893579279476997621651268, 11.42543562020603061177459102481

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