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Properties

Label 2-21e2-63.59-c1-0-23
Degree $2$
Conductor $441$
Sign $0.0148 - 0.999i$
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  + (1.81 + 1.04i)2-s + (0.769 + 1.55i)3-s + (1.19 + 2.07i)4-s + 2.08·5-s + (−0.228 + 3.62i)6-s + 0.819i·8-s + (−1.81 + 2.38i)9-s + (3.79 + 2.18i)10-s − 3.22i·11-s + (−2.29 + 3.44i)12-s + (−2.68 − 1.55i)13-s + (1.60 + 3.24i)15-s + (1.53 − 2.65i)16-s + (0.816 − 1.41i)17-s + (−5.79 + 2.43i)18-s + (−4.79 + 2.76i)19-s + ⋯
L(s)  = 1  + (1.28 + 0.740i)2-s + (0.444 + 0.895i)3-s + (0.597 + 1.03i)4-s + 0.934·5-s + (−0.0933 + 1.47i)6-s + 0.289i·8-s + (−0.604 + 0.796i)9-s + (1.19 + 0.692i)10-s − 0.973i·11-s + (−0.661 + 0.995i)12-s + (−0.745 − 0.430i)13-s + (0.415 + 0.837i)15-s + (0.383 − 0.663i)16-s + (0.197 − 0.342i)17-s + (−1.36 + 0.573i)18-s + (−1.09 + 0.634i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0148 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0148 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0148 - 0.999i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.0148 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33116 + 2.29668i\)
\(L(\frac12)\) \(\approx\) \(2.33116 + 2.29668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.769 - 1.55i)T \)
7 \( 1 \)
good2 \( 1 + (-1.81 - 1.04i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 2.08T + 5T^{2} \)
11 \( 1 + 3.22iT - 11T^{2} \)
13 \( 1 + (2.68 + 1.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.816 + 1.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.79 - 2.76i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.16iT - 23T^{2} \)
29 \( 1 + (7.05 - 4.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.16 + 2.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.82 - 4.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.35 + 2.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.974 + 1.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.06 - 7.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.27 - 3.04i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.98 - 3.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.15 - 2.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.336 - 0.583i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.01iT - 71T^{2} \)
73 \( 1 + (-2.96 - 1.71i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.07 + 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.54 + 2.67i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.45 - 4.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.07 - 1.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.39652689953400603817832489582, −10.29992603991993886266981407183, −9.654273513032746487942956871131, −8.557615644511855090807480639550, −7.53851552854179136189447284946, −6.17924716386838757969979274841, −5.60375635479235855150466635271, −4.69549051291155643206802183887, −3.63714880586864585315732100234, −2.55315349050552123841145259835, 1.92305916485085841026313530418, 2.40250081218214115401152774495, 3.86626300491865747562956723252, 5.00891877126938798579107147935, 6.04963828596660481519949868112, 6.89294408919135607721212032620, 8.069512687806419413894517750058, 9.293108691614768613211243011013, 10.09521887675014944738295579986, 11.28264512758476684656375348810

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