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Properties

Label 2-21e2-7.2-c1-0-4
Degree $2$
Conductor $441$
Sign $-0.198 - 0.980i$
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  + (−1.20 + 2.09i)2-s + (−1.91 − 3.31i)4-s + (0.292 − 0.507i)5-s + 4.41·8-s + (0.707 + 1.22i)10-s + (−1 − 1.73i)11-s + 5.41·13-s + (−1.49 + 2.59i)16-s + (3.12 + 5.40i)17-s + (−1.41 + 2.44i)19-s − 2.24·20-s + 4.82·22-s + (1.82 − 3.16i)23-s + (2.32 + 4.03i)25-s + (−6.53 + 11.3i)26-s + ⋯
L(s)  = 1  + (−0.853 + 1.47i)2-s + (−0.957 − 1.65i)4-s + (0.130 − 0.226i)5-s + 1.56·8-s + (0.223 + 0.387i)10-s + (−0.301 − 0.522i)11-s + 1.50·13-s + (−0.374 + 0.649i)16-s + (0.757 + 1.31i)17-s + (−0.324 + 0.561i)19-s − 0.501·20-s + 1.02·22-s + (0.381 − 0.660i)23-s + (0.465 + 0.806i)25-s + (−1.28 + 2.22i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.570154 + 0.696883i\)
\(L(\frac12)\) \(\approx\) \(0.570154 + 0.696883i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 - 2.09i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.292 + 0.507i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.41T + 13T^{2} \)
17 \( 1 + (-3.12 - 5.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.82 + 3.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 + (-3.41 - 5.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.41 + 5.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.87 - 3.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.82 + 4.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (-2.94 - 5.10i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.17 - 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 + (2.87 - 4.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.41T + 97T^{2} \)
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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

−10.91481010657804171757670767456, −10.32379539415796885265354805477, −9.174159289428214464672201417695, −8.421095998918985097017326143838, −7.976255430613453068367264961153, −6.64576961261498520870170667197, −6.03515780160396173200725514791, −5.13875353840431494069549922739, −3.57959267739840412991542623963, −1.20179338033005915762466423335, 0.988486219388395003426174901067, 2.46409907578077639279917822344, 3.42041783992903870632109576289, 4.70112146123794961506625974038, 6.23243657039725400434555352424, 7.54788014737223865923123237174, 8.465286718846239026331305374972, 9.343153394500798950796081205312, 10.03156739398046234621797885932, 10.88493006425781407196551898435

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