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Properties

Label 2-21e2-21.5-c1-0-13
Degree $2$
Conductor $441$
Sign $-0.239 + 0.970i$
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  + (0.358 + 0.207i)2-s + (−0.914 − 1.58i)4-s + (1.46 − 2.53i)5-s − 1.58i·8-s + (1.05 − 0.606i)10-s + (−4.18 + 2.41i)11-s − 2.93i·13-s + (−1.49 + 2.59i)16-s + (−3.53 − 6.12i)17-s + (5.07 + 2.93i)19-s − 5.35·20-s − 2·22-s + (−1.73 − i)23-s + (−1.79 − 3.10i)25-s + (0.606 − 1.05i)26-s + ⋯
L(s)  = 1  + (0.253 + 0.146i)2-s + (−0.457 − 0.791i)4-s + (0.655 − 1.13i)5-s − 0.560i·8-s + (0.332 − 0.191i)10-s + (−1.26 + 0.727i)11-s − 0.812i·13-s + (−0.374 + 0.649i)16-s + (−0.857 − 1.48i)17-s + (1.16 + 0.672i)19-s − 1.19·20-s − 0.426·22-s + (−0.361 − 0.208i)23-s + (−0.358 − 0.621i)25-s + (0.119 − 0.206i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.239 + 0.970i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.239 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816531 - 1.04266i\)
\(L(\frac12)\) \(\approx\) \(0.816531 - 1.04266i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (-0.358 - 0.207i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.46 + 2.53i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.18 - 2.41i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 + (3.53 + 6.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.07 - 2.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 + i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.828iT - 29T^{2} \)
31 \( 1 + (-5.07 + 2.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.70 + 4.68i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.21T + 41T^{2} \)
43 \( 1 - 4.48T + 43T^{2} \)
47 \( 1 + (-2.93 + 5.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.12 + 3.53i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.93 - 5.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.05 - 0.606i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.24 - 7.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.828iT - 71T^{2} \)
73 \( 1 + (6.12 - 3.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.828 - 1.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (5.60 - 9.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.07iT - 97T^{2} \)
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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

−10.60361402995309862350397511686, −9.821100136861252057620402416261, −9.296875784340510315858281129680, −8.206190634134645804129194472507, −7.12346819125449354926718235833, −5.62594894229661328149458533846, −5.27902670969183479244623155539, −4.36352493188811414334768671311, −2.42507246098788218733391352334, −0.77944045081183681427022535563, 2.38495939246558096138989141635, 3.22539349790862693523992195904, 4.48271455413989667283880647095, 5.72680874682413578878749520892, 6.70127431565571720214370875844, 7.75118253096529330239680263440, 8.606284834465564339475635518863, 9.646461445321127115969648129988, 10.65176128430962318881717768900, 11.25624420217130310947073193896

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