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Properties

Label 2-21e2-49.36-c1-0-13
Degree $2$
Conductor $441$
Sign $0.973 - 0.228i$
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  + (0.0515 + 0.225i)2-s + (1.75 − 0.844i)4-s + (2.14 + 2.69i)5-s + (1.10 − 2.40i)7-s + (0.569 + 0.714i)8-s + (−0.497 + 0.624i)10-s + (−0.807 − 3.53i)11-s + (1.14 + 5.01i)13-s + (0.599 + 0.124i)14-s + (2.29 − 2.87i)16-s + (−1.92 − 0.926i)17-s − 6.76·19-s + (6.04 + 2.91i)20-s + (0.757 − 0.364i)22-s + (0.654 − 0.315i)23-s + ⋯
L(s)  = 1  + (0.0364 + 0.159i)2-s + (0.876 − 0.422i)4-s + (0.961 + 1.20i)5-s + (0.416 − 0.909i)7-s + (0.201 + 0.252i)8-s + (−0.157 + 0.197i)10-s + (−0.243 − 1.06i)11-s + (0.317 + 1.39i)13-s + (0.160 + 0.0333i)14-s + (0.573 − 0.719i)16-s + (−0.466 − 0.224i)17-s − 1.55·19-s + (1.35 + 0.651i)20-s + (0.161 − 0.0777i)22-s + (0.136 − 0.0657i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.973 - 0.228i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.973 - 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00727 + 0.232550i\)
\(L(\frac12)\) \(\approx\) \(2.00727 + 0.232550i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 + 2.40i)T \)
good2 \( 1 + (-0.0515 - 0.225i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (-2.14 - 2.69i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (0.807 + 3.53i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-1.14 - 5.01i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (1.92 + 0.926i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 + 6.76T + 19T^{2} \)
23 \( 1 + (-0.654 + 0.315i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (3.41 + 1.64i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 0.282T + 31T^{2} \)
37 \( 1 + (-4.74 - 2.28i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-7.45 - 9.35i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (2.39 - 3.00i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (2.03 + 8.92i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-2.85 + 1.37i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (1.56 - 1.96i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-0.786 - 0.378i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + 2.70T + 67T^{2} \)
71 \( 1 + (11.6 - 5.60i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-0.463 + 2.03i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + (-1.30 + 5.71i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (3.40 - 14.9i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + 7.92T + 97T^{2} \)
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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

−11.09867224809446767547084605846, −10.48454053701676861377607768861, −9.610361638562482676000671982340, −8.315894340977567096925468210575, −7.09594433470598623981661144876, −6.51626940729541130972143223760, −5.85300241769940276820305332980, −4.29714171504833929970766471412, −2.79687959569004266226357160529, −1.73418223476004057837090849689, 1.71619952607674754633580728521, 2.54961397067537317951637545550, 4.34331094324025261249408336902, 5.47572498082838889290028815378, 6.15982431524818910972712002585, 7.53964757393278043145692626680, 8.451588974620905997974361137643, 9.176223572970050664694243531724, 10.30103026747811082938115473732, 11.03108618079405960418939129743

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