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Properties

Label 2-21e2-63.59-c1-0-0
Degree $2$
Conductor $441$
Sign $-0.635 - 0.771i$
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  + (−1.58 − 0.916i)2-s + (0.108 + 1.72i)3-s + (0.678 + 1.17i)4-s + 0.645·5-s + (1.41 − 2.84i)6-s + 1.17i·8-s + (−2.97 + 0.376i)9-s + (−1.02 − 0.591i)10-s + 5.31i·11-s + (−1.95 + 1.30i)12-s + (−4.44 − 2.56i)13-s + (0.0702 + 1.11i)15-s + (2.43 − 4.22i)16-s + (0.814 − 1.41i)17-s + (5.06 + 2.12i)18-s + (2.09 − 1.20i)19-s + ⋯
L(s)  = 1  + (−1.12 − 0.647i)2-s + (0.0628 + 0.998i)3-s + (0.339 + 0.587i)4-s + 0.288·5-s + (0.575 − 1.16i)6-s + 0.416i·8-s + (−0.992 + 0.125i)9-s + (−0.323 − 0.187i)10-s + 1.60i·11-s + (−0.564 + 0.375i)12-s + (−1.23 − 0.711i)13-s + (0.0181 + 0.288i)15-s + (0.609 − 1.05i)16-s + (0.197 − 0.342i)17-s + (1.19 + 0.501i)18-s + (0.479 − 0.276i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.635 - 0.771i$
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: \((2,\ 441,\ (\ :1/2),\ -0.635 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.163629 + 0.346886i\)
\(L(\frac12)\) \(\approx\) \(0.163629 + 0.346886i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.108 - 1.72i)T \)
7 \( 1 \)
good2 \( 1 + (1.58 + 0.916i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 0.645T + 5T^{2} \)
11 \( 1 - 5.31iT - 11T^{2} \)
13 \( 1 + (4.44 + 2.56i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.814 + 1.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.09 + 1.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.47iT - 23T^{2} \)
29 \( 1 + (6.43 - 3.71i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.90 - 2.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.99 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.99 - 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.51 + 2.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.54 + 2.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.04 + 1.18i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.47 + 2.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.18 + 5.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.07 - 8.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 + (-10.2 - 5.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.48 - 6.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.51 + 6.09i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.16 - 3.74i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.3 + 8.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.20038571365379500309632952699, −10.03734862781604429620390511579, −9.873801869536315778114203585090, −9.226061389219092899970609528497, −8.041791076784623265110341182382, −7.21910408898528331544709575021, −5.42354722955034386974523769105, −4.77079670285933070682719027831, −3.13306434507316272073868288015, −1.96184507218361992014426011953, 0.32811359574174062899742284087, 1.98794897954877355470174199744, 3.61863058502096675659405881199, 5.64546096234509099078491114326, 6.29399657734246686051530539383, 7.44521182214859165910978554535, 7.85143779936475799945350551064, 8.948820201277785177358573769628, 9.455455127617136398188473427243, 10.68294927735946629048166623200

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