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Properties

Label 2-21e2-63.38-c1-0-9
Degree $2$
Conductor $441$
Sign $-0.935 + 0.352i$
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  + 1.17i·2-s + (−0.278 + 1.70i)3-s + 0.611·4-s + (−2.16 + 3.75i)5-s + (−2.01 − 0.327i)6-s + 3.07i·8-s + (−2.84 − 0.950i)9-s + (−4.42 − 2.55i)10-s + (1.87 − 1.08i)11-s + (−0.170 + 1.04i)12-s + (2.25 − 1.30i)13-s + (−5.81 − 4.74i)15-s − 2.40·16-s + (0.585 − 1.01i)17-s + (1.12 − 3.35i)18-s + (2.09 − 1.20i)19-s + ⋯
L(s)  = 1  + 0.833i·2-s + (−0.160 + 0.987i)3-s + 0.305·4-s + (−0.968 + 1.67i)5-s + (−0.822 − 0.133i)6-s + 1.08i·8-s + (−0.948 − 0.316i)9-s + (−1.39 − 0.807i)10-s + (0.564 − 0.325i)11-s + (−0.0491 + 0.301i)12-s + (0.624 − 0.360i)13-s + (−1.50 − 1.22i)15-s − 0.600·16-s + (0.142 − 0.245i)17-s + (0.264 − 0.790i)18-s + (0.480 − 0.277i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.935 + 0.352i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.935 + 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218893 - 1.20104i\)
\(L(\frac12)\) \(\approx\) \(0.218893 - 1.20104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.278 - 1.70i)T \)
7 \( 1 \)
good2 \( 1 - 1.17iT - 2T^{2} \)
5 \( 1 + (2.16 - 3.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.87 + 1.08i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.25 + 1.30i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.585 + 1.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.09 + 1.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.16 - 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.589 - 0.340i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.55iT - 31T^{2} \)
37 \( 1 + (-2.55 - 4.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.68 + 6.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.12 - 3.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.14T + 47T^{2} \)
53 \( 1 + (-2.79 - 1.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.83T + 59T^{2} \)
61 \( 1 + 7.17iT - 61T^{2} \)
67 \( 1 - 6.65T + 67T^{2} \)
71 \( 1 + 1.95iT - 71T^{2} \)
73 \( 1 + (-10.3 - 5.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.75T + 79T^{2} \)
83 \( 1 + (-0.796 + 1.37i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.04 - 5.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.36 + 1.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.29142230966154641182142855603, −10.93047827249491972043600629843, −10.01393291320758015055962783524, −8.694191109657015051713122986231, −7.82779393268142757808784961407, −6.85088061344394772085302765765, −6.25191743450550322303825438261, −5.05638479582782536208718405051, −3.58865103752626384588984452466, −2.94815383768868819191359159572, 0.823315667255253450152946719096, 1.77047790530386158244161513379, 3.49617227999570851105774339416, 4.56466045572284913750274199741, 5.88666767086695888889429005630, 7.02763791809351488280749091285, 7.937150261373153704611136780261, 8.758500069313522678957153732335, 9.667220230194020756922241909679, 11.16191875402749023748733518916

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