Properties

Label 2-21e2-63.38-c1-0-13
Degree $2$
Conductor $441$
Sign $-0.771 - 0.636i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.48i·2-s + (0.995 + 1.41i)3-s − 0.203·4-s + (0.154 − 0.267i)5-s + (−2.10 + 1.47i)6-s + 2.66i·8-s + (−1.01 + 2.82i)9-s + (0.396 + 0.228i)10-s + (2.73 − 1.58i)11-s + (−0.202 − 0.288i)12-s + (−3.00 + 1.73i)13-s + (0.532 − 0.0472i)15-s − 4.36·16-s + (2.44 − 4.22i)17-s + (−4.18 − 1.51i)18-s + (−4.62 + 2.67i)19-s + ⋯
L(s)  = 1  + 1.04i·2-s + (0.574 + 0.818i)3-s − 0.101·4-s + (0.0689 − 0.119i)5-s + (−0.859 + 0.603i)6-s + 0.942i·8-s + (−0.339 + 0.940i)9-s + (0.125 + 0.0723i)10-s + (0.825 − 0.476i)11-s + (−0.0585 − 0.0833i)12-s + (−0.833 + 0.481i)13-s + (0.137 − 0.0121i)15-s − 1.09·16-s + (0.592 − 1.02i)17-s + (−0.987 − 0.356i)18-s + (−1.06 + 0.612i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.771 - 0.636i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615344 + 1.71373i\)
\(L(\frac12)\) \(\approx\) \(0.615344 + 1.71373i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.995 - 1.41i)T \)
7 \( 1 \)
good2 \( 1 - 1.48iT - 2T^{2} \)
5 \( 1 + (-0.154 + 0.267i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.73 + 1.58i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.00 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.44 + 4.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.62 - 2.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.17 - 2.98i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.70 - 1.56i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.52iT - 31T^{2} \)
37 \( 1 + (5.92 + 10.2i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.58 + 4.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.75 + 4.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.46T + 47T^{2} \)
53 \( 1 + (-0.0740 - 0.0427i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.08T + 59T^{2} \)
61 \( 1 - 5.42iT - 61T^{2} \)
67 \( 1 + 0.110T + 67T^{2} \)
71 \( 1 + 7.78iT - 71T^{2} \)
73 \( 1 + (-8.32 - 4.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.13T + 79T^{2} \)
83 \( 1 + (-4.42 + 7.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.936 - 1.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 6.34i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.32982652847812880745019888082, −10.50937790393824992906011432286, −9.205488127792750649773304945695, −8.921857425092448043956969839485, −7.67408481321223575181662816648, −7.01523734183642160016343362408, −5.73464713965195876119787950968, −4.94518589950074050950532841792, −3.70865753668393267614157686520, −2.30091411769909713622831730025, 1.18751243821663121730364149254, 2.40351204247986817930669872260, 3.29435263514382448121178623155, 4.60440007369464588118513131365, 6.47820980451605270028134896923, 6.88043146481065205250118896258, 8.169729722120761462521831526764, 9.043694630506872703025161998600, 10.07410535720450926202198851096, 10.74321103990222042372269488460

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