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Properties

Label 2-21e2-441.4-c1-0-25
Degree $2$
Conductor $441$
Sign $0.833 - 0.552i$
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  + (−0.209 + 2.79i)2-s + (−0.318 − 1.70i)3-s + (−5.79 − 0.873i)4-s + (−0.429 + 1.87i)5-s + (4.82 − 0.533i)6-s + (0.156 − 2.64i)7-s + (2.40 − 10.5i)8-s + (−2.79 + 1.08i)9-s + (−5.16 − 1.59i)10-s + (1.10 − 0.531i)11-s + (0.358 + 10.1i)12-s + (−0.0375 + 0.500i)13-s + (7.35 + 0.992i)14-s + (3.33 + 0.131i)15-s + (17.8 + 5.49i)16-s + (5.37 − 0.810i)17-s + ⋯
L(s)  = 1  + (−0.148 + 1.97i)2-s + (−0.183 − 0.982i)3-s + (−2.89 − 0.436i)4-s + (−0.191 + 0.840i)5-s + (1.97 − 0.217i)6-s + (0.0593 − 0.998i)7-s + (0.851 − 3.73i)8-s + (−0.932 + 0.361i)9-s + (−1.63 − 0.503i)10-s + (0.332 − 0.160i)11-s + (0.103 + 2.92i)12-s + (−0.0104 + 0.138i)13-s + (1.96 + 0.265i)14-s + (0.861 + 0.0340i)15-s + (4.45 + 1.37i)16-s + (1.30 − 0.196i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.833 - 0.552i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.833 - 0.552i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816102 + 0.246132i\)
\(L(\frac12)\) \(\approx\) \(0.816102 + 0.246132i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.318 + 1.70i)T \)
7 \( 1 + (-0.156 + 2.64i)T \)
good2 \( 1 + (0.209 - 2.79i)T + (-1.97 - 0.298i)T^{2} \)
5 \( 1 + (0.429 - 1.87i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-1.10 + 0.531i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.0375 - 0.500i)T + (-12.8 - 1.93i)T^{2} \)
17 \( 1 + (-5.37 + 0.810i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (1.07 + 1.85i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.42 + 6.79i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (0.0305 - 0.0778i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + (4.39 + 7.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.726 - 1.85i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (4.96 + 4.60i)T + (3.06 + 40.8i)T^{2} \)
43 \( 1 + (-0.745 + 0.692i)T + (3.21 - 42.8i)T^{2} \)
47 \( 1 + (-0.296 + 3.95i)T + (-46.4 - 7.00i)T^{2} \)
53 \( 1 + (-1.79 - 4.56i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-4.43 + 4.11i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (-3.13 + 0.473i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (1.57 + 2.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.52 - 1.91i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-4.64 + 3.16i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-3.52 + 6.09i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.0308 + 0.411i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (-0.435 - 5.81i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (5.68 + 9.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.04439224505858499926763547695, −10.11583808175380479839947560600, −8.925423273742723231971399442194, −8.034108978128624021169802908945, −7.17379715211452302282547561967, −6.89973851300718105814637702556, −5.96066089465397551221737681323, −4.86255677338588490654797631891, −3.51821030587719588973997566501, −0.69166544151808454443426919151, 1.35851992022728680481952080849, 2.99908177272538943276818381755, 3.82037589844559096219064861554, 5.04551432233771579365400907675, 5.44249898924014285085338577345, 8.128064481500968336380976628019, 8.876684676237620327125092184027, 9.416491767669361038182471590771, 10.19266978565527487575293969427, 11.09685226644955356729778787806

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