Properties

Label 2-21e2-9.7-c1-0-8
Degree $2$
Conductor $441$
Sign $-0.0816 - 0.996i$
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  + (0.247 − 0.429i)2-s + (−1.37 + 1.05i)3-s + (0.877 + 1.51i)4-s + (1.84 + 3.19i)5-s + (0.109 + 0.851i)6-s + 1.86·8-s + (0.792 − 2.89i)9-s + 1.83·10-s + (0.446 − 0.772i)11-s + (−2.80 − 1.17i)12-s + (0.598 + 1.03i)13-s + (−5.90 − 2.46i)15-s + (−1.29 + 2.23i)16-s + 0.249·17-s + (−1.04 − 1.05i)18-s − 2.80·19-s + ⋯
L(s)  = 1  + (0.175 − 0.303i)2-s + (−0.795 + 0.606i)3-s + (0.438 + 0.759i)4-s + (0.825 + 1.43i)5-s + (0.0447 + 0.347i)6-s + 0.658·8-s + (0.264 − 0.964i)9-s + 0.579·10-s + (0.134 − 0.233i)11-s + (−0.809 − 0.337i)12-s + (0.165 + 0.287i)13-s + (−1.52 − 0.636i)15-s + (−0.323 + 0.559i)16-s + 0.0606·17-s + (−0.246 − 0.249i)18-s − 0.644·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0816 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0816 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0816 - 0.996i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.0816 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01502 + 1.10153i\)
\(L(\frac12)\) \(\approx\) \(1.01502 + 1.10153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.37 - 1.05i)T \)
7 \( 1 \)
good2 \( 1 + (-0.247 + 0.429i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.84 - 3.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.446 + 0.772i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.598 - 1.03i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.249T + 17T^{2} \)
19 \( 1 + 2.80T + 19T^{2} \)
23 \( 1 + (1.23 + 2.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.07 + 3.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.79 + 3.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.73T + 37T^{2} \)
41 \( 1 + (2.39 + 4.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.98 - 8.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.08 + 8.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.88T + 53T^{2} \)
59 \( 1 + (0.906 + 1.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.40 - 9.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.514 + 0.891i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 - 1.83T + 73T^{2} \)
79 \( 1 + (-0.899 + 1.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.16 + 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.40T + 89T^{2} \)
97 \( 1 + (-5.52 + 9.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.33351999271225933462356065823, −10.51959339371228334099079042471, −10.03625931654977398932746899547, −8.790974395333280635065089875301, −7.39269443847993015429546206658, −6.52278905471682625127679768710, −5.92113563595524810073609742450, −4.36226633448583058888473604336, −3.34560794291852927557069697089, −2.21543250087331032586365939137, 1.04989572030583147234040521039, 2.01410289020566243792709939357, 4.58078339347544583296054707708, 5.33920474807743627590497015156, 5.99998303689274576875069346925, 6.88583449384823033540900754065, 8.035808525651127278887563720707, 9.129199072987003165495469736383, 10.10178965160664400426250128499, 10.84280226305239612683995182226

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