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Properties

Label 2-21e2-63.41-c1-0-12
Degree $2$
Conductor $441$
Sign $0.999 + 0.0183i$
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  + (−2.05 + 1.18i)2-s + (−1.27 + 1.16i)3-s + (1.81 − 3.14i)4-s + (−1.71 + 2.97i)5-s + (1.24 − 3.91i)6-s + 3.86i·8-s + (0.271 − 2.98i)9-s − 8.15i·10-s + (0.271 − 0.156i)11-s + (1.35 + 6.14i)12-s + (−5.09 − 2.94i)13-s + (−1.27 − 5.81i)15-s + (−0.958 − 1.65i)16-s + 0.953·17-s + (2.98 + 6.46i)18-s − 1.26i·19-s + ⋯
L(s)  = 1  + (−1.45 + 0.838i)2-s + (−0.738 + 0.674i)3-s + (0.907 − 1.57i)4-s + (−0.768 + 1.33i)5-s + (0.507 − 1.59i)6-s + 1.36i·8-s + (0.0906 − 0.995i)9-s − 2.57i·10-s + (0.0819 − 0.0473i)11-s + (0.389 + 1.77i)12-s + (−1.41 − 0.816i)13-s + (−0.329 − 1.50i)15-s + (−0.239 − 0.414i)16-s + 0.231·17-s + (0.703 + 1.52i)18-s − 0.289i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0183i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.999 + 0.0183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.186687 - 0.00170982i\)
\(L(\frac12)\) \(\approx\) \(0.186687 - 0.00170982i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.27 - 1.16i)T \)
7 \( 1 \)
good2 \( 1 + (2.05 - 1.18i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.71 - 2.97i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.271 + 0.156i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.09 + 2.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.953T + 17T^{2} \)
19 \( 1 + 1.26iT - 19T^{2} \)
23 \( 1 + (5.91 + 3.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.43 + 1.98i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.53 - 2.61i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.37T + 37T^{2} \)
41 \( 1 + (-0.0699 + 0.121i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.44 - 2.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.00 + 1.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.9iT - 53T^{2} \)
59 \( 1 + (-0.824 + 1.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.57 + 1.48i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.934 + 1.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 0.409iT - 73T^{2} \)
79 \( 1 + (5.23 + 9.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.00 + 6.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.11T + 89T^{2} \)
97 \( 1 + (-10.5 + 6.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.62390146840099479711948521349, −10.23625595140703542026766930675, −9.508405455095312399966336510214, −8.210512469023929861222465172187, −7.47728602853838129196586950820, −6.67036020755999804530119657947, −5.88297414588061515512182490796, −4.44086870085031892823846965855, −2.89618746243392368864253794981, −0.25497938544962513645716999618, 1.04180995786286328997917455315, 2.25503974140745057042392067285, 4.22907881914916161230698012666, 5.33817540170186888596273930997, 6.91161523808103345011324390657, 7.86087000655099263129514511750, 8.311663781655513151886201736294, 9.467215710578087971156847348170, 10.07716729406972336649182180779, 11.31712092231513277094326396258

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