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Properties

Label 2-21e2-7.2-c1-0-7
Degree $2$
Conductor $441$
Sign $0.605 - 0.795i$
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 + 1.73i)4-s + 7·13-s + (−1.99 + 3.46i)16-s + (−3.5 + 6.06i)19-s + (2.5 + 4.33i)25-s + (−3.5 − 6.06i)31-s + (0.5 − 0.866i)37-s + 5·43-s + (7 + 12.1i)52-s + (7 − 12.1i)61-s − 7.99·64-s + (−5.5 − 9.52i)67-s + (−3.5 − 6.06i)73-s − 14·76-s + (6.5 − 11.2i)79-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + 1.94·13-s + (−0.499 + 0.866i)16-s + (−0.802 + 1.39i)19-s + (0.5 + 0.866i)25-s + (−0.628 − 1.08i)31-s + (0.0821 − 0.142i)37-s + 0.762·43-s + (0.970 + 1.68i)52-s + (0.896 − 1.55i)61-s − 0.999·64-s + (−0.671 − 1.16i)67-s + (−0.409 − 0.709i)73-s − 1.60·76-s + (0.731 − 1.26i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43315 + 0.710455i\)
\(L(\frac12)\) \(\approx\) \(1.43315 + 0.710455i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 7T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−11.12860371615261467851924439168, −10.70602380960564718246505483983, −9.283608278934972193511150123357, −8.397336184619730793573480375057, −7.72728847174516648442637002954, −6.55084563165038723035480240860, −5.79882880499407263774288320665, −4.09192514111419180474122829763, −3.36028575022800212308720570501, −1.78678407936487485488284490821, 1.15061253936147325347589354896, 2.64949038910737600429711607958, 4.15021255882218326662270793415, 5.39149127483252496338818142677, 6.33877706044315874331776533094, 7.00540619073477057371705526330, 8.467170440399022191272434916402, 9.103973899798186914167211511003, 10.36540667070145818476042774431, 10.90353487336046868288197811939

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