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Properties

Label 2-21e2-9.4-c1-0-12
Degree $2$
Conductor $441$
Sign $-0.965 + 0.259i$
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  + (1.35 + 2.35i)2-s + (0.521 + 1.65i)3-s + (−2.68 + 4.65i)4-s + (0.793 − 1.37i)5-s + (−3.17 + 3.46i)6-s − 9.15·8-s + (−2.45 + 1.72i)9-s + 4.30·10-s + (0.674 + 1.16i)11-s + (−9.08 − 2.00i)12-s + (1.58 − 2.75i)13-s + (2.68 + 0.593i)15-s + (−7.05 − 12.2i)16-s + 2.80·17-s + (−7.38 − 3.43i)18-s + 0.625·19-s + ⋯
L(s)  = 1  + (0.959 + 1.66i)2-s + (0.301 + 0.953i)3-s + (−1.34 + 2.32i)4-s + (0.354 − 0.614i)5-s + (−1.29 + 1.41i)6-s − 3.23·8-s + (−0.818 + 0.574i)9-s + 1.36·10-s + (0.203 + 0.352i)11-s + (−2.62 − 0.580i)12-s + (0.440 − 0.763i)13-s + (0.692 + 0.153i)15-s + (−1.76 − 3.05i)16-s + 0.679·17-s + (−1.74 − 0.809i)18-s + 0.143·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.965 + 0.259i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.965 + 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.298983 - 2.26306i\)
\(L(\frac12)\) \(\approx\) \(0.298983 - 2.26306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.521 - 1.65i)T \)
7 \( 1 \)
good2 \( 1 + (-1.35 - 2.35i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.793 + 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.674 - 1.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.58 + 2.75i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.80T + 17T^{2} \)
19 \( 1 - 0.625T + 19T^{2} \)
23 \( 1 + (-0.142 + 0.246i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.27 - 3.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.71 - 6.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.02T + 37T^{2} \)
41 \( 1 + (-5.01 + 8.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.12 + 5.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.57 + 9.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.78T + 53T^{2} \)
59 \( 1 + (2.28 - 3.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.192 - 0.333i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.26 + 2.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.45T + 71T^{2} \)
73 \( 1 + 0.468T + 73T^{2} \)
79 \( 1 + (-7.85 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.99 + 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.58T + 89T^{2} \)
97 \( 1 + (-7.22 - 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.98899148405473760626828048854, −10.55504349674956101345836209184, −9.363238081170103434280445292623, −8.705434682206819928799567926424, −7.917217831194494159200302859078, −6.85734586894363843831942269562, −5.52937497684824341681314287648, −5.23754026669783140266905142916, −4.09535831953472570092121507582, −3.17472111563172880649689146930, 1.17172148960976906801942201666, 2.38736828795049813495316335968, 3.22332227948477598607836144015, 4.39531273405875379095870517028, 5.89883645915961494540203159844, 6.40745474383066588549072616612, 7.990722368703889447187080305403, 9.274257802947085208887963737577, 9.885423215426595292637352434503, 11.18682137926909140055936749262

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