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Properties

Label 2-21e2-7.4-c1-0-7
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  + (−1.32 − 2.29i)2-s + (−2.5 + 4.33i)4-s + 7.93·8-s + (2.64 − 4.58i)11-s + (−5.49 − 9.52i)16-s − 14·22-s + (−2.64 − 4.58i)23-s + (2.5 − 4.33i)25-s − 10.5·29-s + (−6.61 + 11.4i)32-s + (−3 − 5.19i)37-s + 12·43-s + (13.2 + 22.9i)44-s + (−7 + 12.1i)46-s − 13.2·50-s + ⋯
L(s)  = 1  + (−0.935 − 1.62i)2-s + (−1.25 + 2.16i)4-s + 2.80·8-s + (0.797 − 1.38i)11-s + (−1.37 − 2.38i)16-s − 2.98·22-s + (−0.551 − 0.955i)23-s + (0.5 − 0.866i)25-s − 1.96·29-s + (−1.16 + 2.02i)32-s + (−0.493 − 0.854i)37-s + 1.82·43-s + (1.99 + 3.45i)44-s + (−1.03 + 1.78i)46-s − 1.87·50-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

\(L(1)\) \(\approx\) \(0.0431640 - 0.680175i\)
\(L(\frac12)\) \(\approx\) \(0.0431640 - 0.680175i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 2.29i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.64 + 4.58i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.64 + 4.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.5T + 29T^{2} \)
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 \( 1 + (-5.29 + 9.16i)T + (-26.5 - 45.8i)T^{2} \)
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 \( 1 + 5.29T + 71T^{2} \)
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

−10.82609173296786453330035239923, −9.911812943306868928725525944038, −8.943031227210958481236175773362, −8.516195764939302457246275916807, −7.37985377558115264145451324182, −5.92317704149626451149910777062, −4.23004013721510196810552384000, −3.38024253899097073966283710723, −2.14009531752356935003789463378, −0.62738357114098857897071554810, 1.57524628942162217482476555325, 4.10954644861899663574982754642, 5.25115218957661253206508273131, 6.15070334950467214322174731646, 7.26599583516057262854233745818, 7.53675227555732028248516407599, 8.892251051294420503655452830306, 9.412375924765196334903300457434, 10.16521409613907406703997374738, 11.30181091589250696727956659177

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