Properties

Label 2-21e2-7.4-c1-0-9
Degree $2$
Conductor $441$
Sign $0.991 - 0.126i$
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  + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + 3·8-s + (2 − 3.46i)11-s + (0.500 + 0.866i)16-s + 3.99·22-s + (4 + 6.92i)23-s + (2.5 − 4.33i)25-s − 2·29-s + (2.50 − 4.33i)32-s + (3 + 5.19i)37-s − 12·43-s + (−1.99 − 3.46i)44-s + (−3.99 + 6.92i)46-s + 5·50-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + 1.06·8-s + (0.603 − 1.04i)11-s + (0.125 + 0.216i)16-s + 0.852·22-s + (0.834 + 1.44i)23-s + (0.5 − 0.866i)25-s − 0.371·29-s + (0.441 − 0.765i)32-s + (0.493 + 0.854i)37-s − 1.82·43-s + (−0.301 − 0.522i)44-s + (−0.589 + 1.02i)46-s + 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92943 + 0.122442i\)
\(L(\frac12)\) \(\approx\) \(1.92943 + 0.122442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5 - 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.18803891444180594550764081904, −10.30471708388913224954778517263, −9.308040957925004218281811246323, −8.287968396041543380977247870036, −7.24377531406218404291946005590, −6.36579893512035772327223491545, −5.58751412495102888234703243537, −4.53634707812790858538234661528, −3.17051580992645947470498223898, −1.36244013126385186572929554582, 1.72308377574922347240824843108, 2.97309757549757489362482940577, 4.13031728033590283530223709279, 5.01970630315614486339063133602, 6.62665534373549917236332618591, 7.29139196053559166721759899932, 8.394894458611223772036190976834, 9.423907224831202791519044739523, 10.42260184288219637129287499863, 11.22118699307712351797105705482

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