Properties

Label 2-21e2-63.4-c1-0-23
Degree $2$
Conductor $441$
Sign $0.940 + 0.338i$
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  + (0.863 − 1.49i)2-s + (0.615 + 1.61i)3-s + (−0.490 − 0.849i)4-s + 3.51·5-s + (2.95 + 0.477i)6-s + 1.75·8-s + (−2.24 + 1.99i)9-s + (3.03 − 5.25i)10-s − 6.09·11-s + (1.07 − 1.31i)12-s + (0.560 − 0.970i)13-s + (2.16 + 5.68i)15-s + (2.49 − 4.32i)16-s + (−0.601 + 1.04i)17-s + (1.04 + 5.07i)18-s + (1.10 + 1.90i)19-s + ⋯
L(s)  = 1  + (0.610 − 1.05i)2-s + (0.355 + 0.934i)3-s + (−0.245 − 0.424i)4-s + 1.57·5-s + (1.20 + 0.195i)6-s + 0.621·8-s + (−0.747 + 0.664i)9-s + (0.958 − 1.66i)10-s − 1.83·11-s + (0.310 − 0.380i)12-s + (0.155 − 0.269i)13-s + (0.557 + 1.46i)15-s + (0.624 − 1.08i)16-s + (−0.146 + 0.252i)17-s + (0.245 + 1.19i)18-s + (0.252 + 0.438i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.940 + 0.338i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.940 + 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.54732 - 0.444760i\)
\(L(\frac12)\) \(\approx\) \(2.54732 - 0.444760i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.615 - 1.61i)T \)
7 \( 1 \)
good2 \( 1 + (-0.863 + 1.49i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3.51T + 5T^{2} \)
11 \( 1 + 6.09T + 11T^{2} \)
13 \( 1 + (-0.560 + 0.970i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.601 - 1.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 - 1.90i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.27T + 23T^{2} \)
29 \( 1 + (3.10 + 5.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.0942 - 0.163i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.78 + 3.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.68 + 2.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.90 + 3.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.86 + 4.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.16 + 7.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.63 - 9.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.00 - 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.95 - 6.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + (-2.65 + 4.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.60 - 7.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.624 - 1.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.77 + 4.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.24 - 14.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.73288917453705242503779429993, −10.32148162433851291920448279015, −9.798693077855227065639838425937, −8.622035088296191058931071915162, −7.54452273311854475696387168488, −5.68121384973146362344717331175, −5.28069269813903961651901903921, −4.01053800003420767045809492943, −2.75811558766357016020762555003, −2.11558620824622515295904710968, 1.74238796257928009120015058200, 2.86232828040963368964357407197, 4.95190536735116681430705264818, 5.65343811665519021900592654854, 6.40562076819071898603102894016, 7.27828840622851972983701289662, 8.097417761749946148875718028097, 9.191788567516140802396830372889, 10.20898022336639234820846954116, 11.10330670164391077135622266775

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