Properties

Label 2-21e2-441.142-c1-0-26
Degree $2$
Conductor $441$
Sign $0.893 - 0.448i$
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  + (−0.0500 + 0.127i)2-s + (−1.68 + 0.411i)3-s + (1.45 + 1.34i)4-s + (2.28 − 1.09i)5-s + (0.0317 − 0.234i)6-s + (2.44 + 1.01i)7-s + (−0.491 + 0.236i)8-s + (2.66 − 1.38i)9-s + (0.0259 + 0.345i)10-s + (−2.32 − 2.90i)11-s + (−2.99 − 1.66i)12-s + (0.926 − 2.36i)13-s + (−0.251 + 0.260i)14-s + (−3.38 + 2.78i)15-s + (0.290 + 3.87i)16-s + (3.88 − 3.60i)17-s + ⋯
L(s)  = 1  + (−0.0353 + 0.0901i)2-s + (−0.971 + 0.237i)3-s + (0.726 + 0.673i)4-s + (1.02 − 0.491i)5-s + (0.0129 − 0.0959i)6-s + (0.922 + 0.385i)7-s + (−0.173 + 0.0836i)8-s + (0.887 − 0.461i)9-s + (0.00819 + 0.109i)10-s + (−0.699 − 0.877i)11-s + (−0.865 − 0.482i)12-s + (0.257 − 0.654i)13-s + (−0.0673 + 0.0695i)14-s + (−0.874 + 0.719i)15-s + (0.0726 + 0.969i)16-s + (0.943 − 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.893 - 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49009 + 0.353121i\)
\(L(\frac12)\) \(\approx\) \(1.49009 + 0.353121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 - 0.411i)T \)
7 \( 1 + (-2.44 - 1.01i)T \)
good2 \( 1 + (0.0500 - 0.127i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-2.28 + 1.09i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (2.32 + 2.90i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.926 + 2.36i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-3.88 + 3.60i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (2.59 - 4.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.816 - 3.57i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-2.55 - 0.788i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-5.40 + 9.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.32 - 1.64i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-2.77 - 1.89i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (8.66 - 5.90i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (1.12 - 2.87i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (12.3 - 3.82i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-6.38 + 4.35i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (6.13 - 5.69i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (4.21 - 7.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.21 + 9.68i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (8.46 + 1.27i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (0.0538 + 0.0932i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.62 + 11.7i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (4.55 + 11.6i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-7.56 + 13.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.34705420820434384364821697292, −10.39714549376624786599096667488, −9.572693277458208465733092298352, −8.226881935069106449033354323860, −7.67172458795550842500712039316, −5.97756173399175661854040419672, −5.83903033204675051655942637656, −4.64185529248964767645980960254, −3.00466173007294673968387341869, −1.47139597408839787811187928800, 1.43790660847704036641824868817, 2.34662914447197479390671801205, 4.60740466565845877550623024732, 5.41807209457562938906425134569, 6.46963013652599087806818635760, 6.92909669409781455851377322190, 8.140804190702682551987208770357, 9.760125390982870154861458052075, 10.46047498138206070983344457363, 10.78148024815514580068683491591

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