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Properties

Label 2-21e2-441.16-c1-0-32
Degree $2$
Conductor $441$
Sign $0.775 + 0.631i$
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  + (−0.216 − 0.0326i)2-s + (1.18 + 1.26i)3-s + (−1.86 − 0.575i)4-s + (−1.19 − 0.576i)5-s + (−0.215 − 0.311i)6-s + (0.0273 − 2.64i)7-s + (0.779 + 0.375i)8-s + (−0.182 + 2.99i)9-s + (0.240 + 0.163i)10-s + (2.70 − 3.38i)11-s + (−1.48 − 3.03i)12-s + (2.80 + 0.422i)13-s + (−0.0922 + 0.571i)14-s + (−0.693 − 2.19i)15-s + (3.06 + 2.09i)16-s + (−0.953 + 0.294i)17-s + ⋯
L(s)  = 1  + (−0.153 − 0.0230i)2-s + (0.685 + 0.728i)3-s + (−0.932 − 0.287i)4-s + (−0.535 − 0.257i)5-s + (−0.0880 − 0.127i)6-s + (0.0103 − 0.999i)7-s + (0.275 + 0.132i)8-s + (−0.0609 + 0.998i)9-s + (0.0759 + 0.0518i)10-s + (0.814 − 1.02i)11-s + (−0.429 − 0.876i)12-s + (0.776 + 0.117i)13-s + (−0.0246 + 0.152i)14-s + (−0.179 − 0.566i)15-s + (0.767 + 0.523i)16-s + (−0.231 + 0.0713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.775 + 0.631i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14881 - 0.408719i\)
\(L(\frac12)\) \(\approx\) \(1.14881 - 0.408719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.18 - 1.26i)T \)
7 \( 1 + (-0.0273 + 2.64i)T \)
good2 \( 1 + (0.216 + 0.0326i)T + (1.91 + 0.589i)T^{2} \)
5 \( 1 + (1.19 + 0.576i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-2.70 + 3.38i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-2.80 - 0.422i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (0.953 - 0.294i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-3.59 + 6.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.999 + 4.37i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-5.99 - 5.55i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (-2.40 + 4.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.17 + 1.09i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-0.615 - 8.21i)T + (-40.5 + 6.11i)T^{2} \)
43 \( 1 + (-0.324 + 4.33i)T + (-42.5 - 6.40i)T^{2} \)
47 \( 1 + (1.87 + 0.282i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (2.03 - 1.88i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (0.904 - 12.0i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-2.93 + 0.906i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (3.23 - 5.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.10 + 9.22i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-2.38 + 6.08i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (5.23 + 9.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.6 + 1.61i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (10.6 - 1.60i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-2.35 + 4.07i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.80971792298749930612799125884, −10.08680335945259900898756184976, −9.003925018254846624955799590920, −8.620474096547849020972462457918, −7.68519831777694946248767309923, −6.30474436506863496378664457043, −4.79954730773086323192562671521, −4.18381294496815075868229214243, −3.26162765764735068234694098226, −0.884299977154685725389040056157, 1.57992109398833863399947573643, 3.24614998436389542297510641534, 4.09419129739750473770741872111, 5.58407232242744996354521406881, 6.75677744857842492755357256284, 7.84974467323585282097103920523, 8.353426279834890793341824668809, 9.334204904614130786505076756219, 9.884383059565887049566559352066, 11.62463980980210907963258691046

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