Properties

Label 2-21e2-9.7-c1-0-20
Degree $2$
Conductor $441$
Sign $0.145 + 0.989i$
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 + (−1.09 − 1.34i)3-s + (−0.490 − 0.849i)4-s + (1.75 + 3.04i)5-s + (−2.95 + 0.477i)6-s + 1.75·8-s + (−0.604 + 2.93i)9-s + 6.06·10-s + (3.04 − 5.27i)11-s + (−0.603 + 1.58i)12-s + (−0.560 − 0.970i)13-s + (2.16 − 5.68i)15-s + (2.49 − 4.32i)16-s − 1.20·17-s + (3.87 + 3.44i)18-s + 2.20·19-s + ⋯
L(s)  = 1  + (0.610 − 1.05i)2-s + (−0.631 − 0.775i)3-s + (−0.245 − 0.424i)4-s + (0.785 + 1.36i)5-s + (−1.20 + 0.195i)6-s + 0.621·8-s + (−0.201 + 0.979i)9-s + 1.91·10-s + (0.918 − 1.59i)11-s + (−0.174 + 0.458i)12-s + (−0.155 − 0.269i)13-s + (0.557 − 1.46i)15-s + (0.624 − 1.08i)16-s − 0.292·17-s + (0.912 + 0.810i)18-s + 0.505·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.145 + 0.989i$
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: \((2,\ 441,\ (\ :1/2),\ 0.145 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47377 - 1.27247i\)
\(L(\frac12)\) \(\approx\) \(1.47377 - 1.27247i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.09 + 1.34i)T \)
7 \( 1 \)
good2 \( 1 + (-0.863 + 1.49i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.75 - 3.04i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.04 + 5.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.560 + 0.970i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.20T + 17T^{2} \)
19 \( 1 - 2.20T + 19T^{2} \)
23 \( 1 + (-0.636 - 1.10i)T + (-11.5 + 19.9i)T^{2} \)
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 \( 1 - 3.57T + 37T^{2} \)
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 \( 1 + 8.33T + 53T^{2} \)
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 \( 1 - 5.31T + 73T^{2} \)
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 \( 1 + 5.54T + 89T^{2} \)
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

−11.19188063779915293322945699487, −10.56239553595318963078334141277, −9.486243315620005212918825585323, −7.997453538562310374609135629075, −6.91580454655112337969043857175, −6.19214629105948211403723969486, −5.23338845719040734275771181074, −3.53016366025602768686240561197, −2.71288455032891739226792739285, −1.44023133169007502344726197898, 1.59396908556739928333579545580, 4.24267524189422533018230118615, 4.68116417725399665322389763288, 5.54883091837417712014075635688, 6.36295857465930100333644576014, 7.31669235234925246665763607370, 8.722110681702255774640696150346, 9.580849740899830427836518446139, 10.09112970343683695127960336706, 11.50059098178130169142426599760

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